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The stability analysis of a 2D Keller–Segel–Navier–Stokes system in fast signal diffusion

Published online by Cambridge University Press:  31 March 2022

MIN LI
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China emails: limin_pde@163.com; zxiang@uestc.edu.cn
ZHAOYIN XIANG
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China emails: limin_pde@163.com; zxiang@uestc.edu.cn
GUANYU ZHOU
Affiliation:
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China email: wind_geno@live.com
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Abstract

This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.

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Papers
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

Keller–Segel system and fast signal diffusion limit. Chemotaxis, the biased movement of cells along spatial gradients of chemical cues, plays an important role in numerous biological circumstances such as bacterial aggregation, spatial pattern formation, embryonic morphogenesis, immune response and also tumour-induced angiogenesis. The most basic mathematical model for chemotaxis was originally derived in 1953 by Patlak [Reference Patlak25] and then in 1970 by Keller and Segel [Reference Keller and Segel13]. The main unknowns in this so-called Keller–Segel model are the nonnegative cell density n and chemical concentration c, which satisfy the parabolic–parabolic reaction-diffusion equations:

(1.1) \begin{equation} \begin{cases} \partial_tn = \tau_1 \Delta n - \nabla \cdot \big(n S \nabla c\big),& x\in \Omega, \, t>0, \\ \\[-12pt] \partial_tc = \tau_2 \Delta c - c + n, & x\in \Omega, \, t>0, \end{cases} \end{equation}

where $\tau_1$ is the positive cell diffusivity and $\tau_2$ stands for the positive diffusivity of the chemical. In many realistic modelling situations, the chemotactic sensitivity S has to be allowed to depend on the cell density n and on the chemical concentration c.

The celebrated Keller–Segel system (1.1) has been well-studied with regard to biological implications, but beyond this, during the last decades quite a thorough comprehension of its mathematical features has grown in various directions. For instance, a striking feature of system (1.1) appears to be the occurrence of some solutions blowing up in finite time ([Reference Herrero and Velázquez11]), which is commonly viewed as mathematically expressing numerous processes of spontaneous cell aggregation which can be observed in experiments. In the spatially two-dimensional framework, in particular, it was shown in [Reference Nagai, Senba and Yoshida24, Reference Herrero and Velázquez11] that system (1.1) possesses some solutions which blow up in finite time provided that the initially present total mass $\int_{\Omega}n(x, 0)$ is large enough, whereas solutions remain bounded whenever $\int_{\Omega}n(x, 0)$ is small; as a precise value distinguishing the respective mass regimes either allowing for or suppressing explosions, the critical mass $m_c=8\pi$ could be identified in the spatially radial setting or $\Omega=\mathbb{R}^2$ . Such explosion phenomena can be ruled out when S is related to the prototypical assumption of volume-filling effect. Precisely, it has been shown in [Reference Horstmann and Winkler12] that for the two-dimensional no-flux boundary value problem of system (1.1) with n-dependent sensitivities S(n), all solutions are global and uniformly bounded provided that

(1.2) \begin{equation}S(n)\le \frac{C_S}{(1+n)^{\alpha}} \qquad\textrm{ with} \qquad \alpha>0\end{equation}

for some positive constant $C_S$ , while the solution may blow up if $\Omega \subset \mathbb{R}^2$ is a ball and

\begin{equation*}S(n)\ge \frac{C_S}{(1+n)^{\alpha}} \qquad\textrm{ with\,\,} \qquad\alpha<0.\end{equation*}

Due to the experimental facts, the diffusion coefficient $\tau_2$ of the chemoattractant c is usually assumed to be large and the ratio between the diffusivity of the cells and of the chemoattractant

\begin{align*} \\[-28pt] \epsilon:=\frac{\tau_1}{\tau_2}\end{align*}

can be regarded as a relaxation time scale such that $\epsilon^{-1}$ is the rate towards equilibrium. Then taking into account the different time scales of the two diffusion processes and replacing $\tau_1 t$ with t in the original parabolic–parabolic (PP) system (1.1), we obtain

(1.3) \begin{equation} \left\{ \begin{split} & \,\, \partial_t n = \Delta n - \frac{1}{\tau_1} \nabla \cdot \big(n S(n) \nabla c\big), \\ \\[-8pt] & \epsilon \partial_t c = \Delta c - \frac{1}{\tau_2} c + \frac{1}{\tau_2} n. \end{split} \right.\end{equation}

The formal choice $\epsilon=0$ in (1.3) will lead to a corresponding parabolic–elliptic (PE) system:

(1.4) \begin{equation} \left\{ \begin{split} & \partial_t n = \Delta n - \frac{1}{\tau_1} \nabla \cdot \big(n S(n) \nabla c\big), \\ \\[-8pt] & \,\,\,\,\, 0 = \Delta c - \frac{1}{\tau_2} c + \frac{1}{\tau_2} n, \end{split} \right.\end{equation}

which describes the chemical concentration evolution in a quasi-stationary approximation. The PE system substantially differs from its fully PP system due to the circumstance that the former cross-diffusive interaction involves a certain memory. The theory of the PE system (1.4) is relatively well developed. For instance, a comprehensive picture was obtained in [Reference Senba and Suzuki27] for the two-dimensional PE system (1.4): the Dirac mass formation and finiteness of blow-up points were derived without substantial restrictions. Even in the 2D mass critical case, in which solutions to the Cauchy problem of the minimal PE system (1.4) exist globally but blow up in infinite time, it is known that the spatial profile near the corresponding blow-up time $T=\infty$ is essentially dictated by Dirac distributions (see [Reference Blanchet, Carrillo and Masmoudi3, Reference Ghoul and Masmoudi9]).

Several nice analytical results in [Reference Biler and Brandolese2, Reference Kurokiba and Ogawa17, Reference Lemarié-Rieusset18, Reference Raczynski26] showed the stability properties of solutions to Keller–Segel system in the whole space $\mathbb{R}^2$ as $\epsilon\to 0$ : solutions of the PP system (1.3) converge in some special cases (e.g. for $c(x, 0)\equiv 0$ , for some finite time T or for small initial data) to those of the PE system (1.4) (see also [Reference Freitag8, Reference Mizukami23] for the initial-boundary value problem). These partially solved an old question raised by Biler [Reference Biler1]. Recently, Liu et al. [Reference Liu, Wang and Zhou20] proposed a semi-discrete scheme based on a symmetrisation reformation and showed that their new scheme is stable as $\epsilon\to 0$ provided that the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime.

Keller–Segel–(Navier–)Stokes system and fast signal diffusion limit. Partially motivated by the striking experiments in [Reference Tuval, Cisneros, Dombrowski, Wolgemuth, Kessler and Goldstein28], the typical models describing the interaction between populations of chemotactically migrating individuals and viscous fluid environments have become the best-studied models in mathematical biology (see [Reference Duan, Lorz and Markowich6, Reference Winkler36]). In [Reference Kiselev and Ryzhik14, Reference Kiselev and Ryzhik15], Kiselev and Ryzhik considered the effect chemotactic attraction on reproduction of some invertebrates, such as sea urchins, anemones and corals. In particular, they investigated the phenomenon of broadcast spawning whereby males and females release sperm and egg gametes into the surrounding flow. For the coral spawning problem, there is experimental evidence that eggs release a chemical that attracts sperm (see [Reference Coll, Bowden, Meehan, Konig, Carroll, Tapiolas, Alino, Heaton, de Nys, Leone, Maida, Aceret, Willis, Babcock, Willis, Florian, Clayton and Miller4, Reference Coll, Leone, Bowden, Carroll, Konig, Heaton, de Nys, Maida, Alino, Willis, Babcock, Florian, Clayton, Miller and Alderslade5]). This leads us to investigate the PP-fluid model:

(1.5) \begin{eqnarray} \left\{ \begin{array}{llc} \,\, \partial_tn+u\cdot\nabla n=\Delta n-\nabla\cdot \big( nS\nabla c\big),\\ \\[-7pt] \epsilon \partial_tc+u\cdot\nabla c \, \, = \Delta c - c +n, \\ \\[-7pt] \,\, \partial_tu +\kappa (u\cdot \nabla)u +\nabla P = \Delta u + n\nabla\phi , \\ \\[-7pt]\qquad \quad \,\,\, \, \nabla\cdot u=0 \end{array} \right.\end{eqnarray}

and to consider the effect of the surrounding fluid on the chemotaxis, where the additional unknowns are the fluid velocity u and the associated pressure P. Here the given potential function $\phi=\phi(x, t)$ arose from the chemotactic boycott effect and the coefficient $\kappa\in \mathbb{R}$ measured the strength of nonlinear fluid convection.

Due to the possible singularity in the fluid-free case as mentioned before, it is natural to require a volume-filling hypothesis of the form (1.2) to establish global solvability for the PP-fluid system (1.5) and its variants (see [Reference Wang, Winkler and Xiang29, Reference Wang and Xiang34, Reference Winkler37, Reference Zheng41]). For $\kappa=1$ , in particular, it has been revealed in [Reference Wang, Winkler and Xiang29] that under the rotational chemotactic assumption of the form $S=S(x, n, c)$ (see [Reference Xue and Othmer39]) satisfying the natural volume-filling hypothesis

(1.6) \begin{equation} |S(x,n,c)|\le \frac{C_S}{(1+n)^{\alpha}} \end{equation}

with some positive constant $C_S$ , there exist global bounded classical solutions to the 2D homogeneous Neumann–Neumann–Dirichlet initial-boundary value problem of system (1.5) whenever $\alpha>0$ , which is accurately consistent with the case of fluid-free system (1.1). A corresponding 3D setting possesses a globally defined weak solution whenever $\alpha>\frac13$ (see [Reference Ke and Zheng16]). For $\kappa=0$ , on the other hand, Lorz [Reference Lorz21] illustrated the behaviour of the 2D PE-fluid system

(1.7) \begin{eqnarray} \left\{ \begin{array}{llc} \,\, \partial_tn+u\cdot\nabla n=\Delta n-\nabla\cdot \big( nS\nabla c\big),\\ \\[-7pt] \qquad \quad u\cdot\nabla c \, \, = \Delta c - c +n, \\ \\[-7pt] \,\, \partial_tu +\kappa (u\cdot \nabla)u +\nabla P = \Delta u + n\nabla\phi , \\ \\[-7pt] \qquad \quad \,\,\,\, \nabla\cdot u=0 \end{array} \right.\end{eqnarray}

(i.e. $\epsilon=0$ in (1.5)) with different numerical examples and in particular gave the numerical evidence that above the critical mass of $8\pi$ solutions still exist for PE-fluid system (1.7). Recently, Zheng [Reference Zheng42] proved that if $\alpha>0$ , then the associated initial-boundary-value problem (1.7) possesses a global bounded classical solution for any sufficiently smooth initial data $(n_0, u_0)$ satisfying some compatibility conditions. The Dirichlet boundary effects for the signal have also been investigated in [Reference Wang, Winkler and Xiang31, Reference Wang, Winkler and Xiang32, Reference Wang, Winkler and Xiang33].

In the last 2 years, two rigorous stability analyses for the chemotaxis-fluid system have been done by [Reference Wang, Winkler and Xiang30, Reference Li and Xiang19]. In particular, Wang et al. [Reference Wang, Winkler and Xiang30] affirmed that under some assumptions on the model ingredients, that is,

\begin{equation*}\sup_{\epsilon} \|\nabla c_{\epsilon}\|_{L^\lambda((0,T);L^q(\Omega))} < \infty\qquad\mathrm{and}\qquad\sup_{\epsilon} \|u_\epsilon\|_{L^\infty((0,T);L^r(\Omega))} < \infty\end{equation*}

with some $\lambda\in (2,\infty]$ , $q>d$ and $r>\max\big\{2,d\big\}$ fulfilling $\frac{1}{\lambda} + \frac{d}{2q} < \frac{1}{2}$ , there exists a subsequence for solutions $(n_{\epsilon}, c_{\epsilon}, u_\epsilon)$ to the initial-boundary value problem of the fully PP-fluid system

(1.8) \begin{equation}\left\{\begin{split} & \, \, \partial_t n_{\epsilon} + u_\epsilon\cdot\nabla n_\epsilon=\Delta n_\epsilon-\nabla\cdot\big(n_\epsilon S(x,n_{\epsilon},c_{\epsilon})\nabla c_\epsilon\big)+f(x,n_{\epsilon},c_{\epsilon}), \\ \\[-7pt] &\epsilon \partial_t c_{\epsilon}+ u_\epsilon\cdot\nabla c_\epsilon \, =\Delta c_\epsilon-c_\epsilon+n_\epsilon, \\ \\[-7pt] & \,\, \partial_t u_{\epsilon } +\kappa (u_\epsilon\cdot\nabla)u_\epsilon +\nabla P_\epsilon = \Delta u_\epsilon + n_\epsilon\nabla\phi, \\ \\[-7pt] & \qquad \qquad \,\, \nabla\cdot u_\epsilon=0\end{split}\right.\end{equation}

converging to the solution of its PE-fluid counterpart

\begin{equation*} \left\{ \begin{split}& \partial_tn+u\cdot\nabla n=\Delta n-\nabla\cdot \big( nS(x, n, c) \nabla c\big) + f(x,n, c),\\\\[-7pt] &\qquad\quad u\cdot\nabla c=\Delta c - c + n, \\\\[-7pt] & \partial_tu + \kappa (u\cdot\nabla)u + \nabla P = \Delta u + n\nabla\phi, \\ \\[-7pt] & \qquad \quad \,\, \nabla\cdot u=0 \end{split} \right.\end{equation*}

in $\Omega\times(0, T)$ as $\epsilon\to 0$ , where $\kappa \in \mathbb{R}$ and $\Omega\subset\mathbb R^d$ ( $d\ge1$ ) is a smoothly bounded convex domain. Then under the volume-filling assumption (1.6) with $\alpha>0$ , the first two authors [Reference Li and Xiang19] established an algebraic convergence rate of the fast signal diffusion limit for the PP-Stokes system (i.e., $\kappa=0$ in (1.8)) with $f\equiv0$ and general large initial data and removed the restriction to asserting convergence only along some subsequence in [Reference Wang, Winkler and Xiang30].

Main results. In the present work, we will further consider the stability in a full Keller–Segel–Navier–Stokes system. Precisely, we investigate the convergence of solutions of the PP-fluid system

(1.9) \begin{equation}\left\{\begin{split} & \, \partial_t n_{\epsilon }+u_\epsilon\cdot\nabla n_\epsilon=\Delta n_\epsilon - \nabla\cdot \big(n_{\epsilon} S(x,n_{\epsilon},c_{\epsilon}) \nabla c_\epsilon\big), & \qquad x\in\Omega,\,\, t>0,\\ \\[-12pt] &\epsilon\partial_t c_{\epsilon} + u_\epsilon\cdot\nabla c_\epsilon= \Delta c_\epsilon-c_\epsilon+n_\epsilon,& \qquad x\in\Omega,\,\, t>0,\\ \\[-12pt] & \, \partial_t u_{\epsilon } +(u_\epsilon\cdot\nabla)u_\epsilon + \nabla P_\epsilon = \Delta u_\epsilon+n_\epsilon\nabla\phi, & \qquad x\in\Omega,\,\, t>0,\\\\[-12pt] & \qquad \qquad \nabla\cdot u_\epsilon=0, & \qquad x\in\Omega,\,\, t>0,\\ \\[-12pt] &\big(\nabla n_{\epsilon}-n_{\epsilon} S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\big) \cdot\nu=\nabla c_\epsilon\cdot \nu=0, \quad u_\epsilon=0, & \qquad x\in\partial\Omega,\,\, t>0,\\ \\[-12pt] &n_\epsilon(x,0)=n_0(x),\quad c_\epsilon(x,0)=c_0(x),\quad u_\epsilon(x,0)=u_0(x), & \qquad x\in\Omega\end{split}\right.\end{equation}

to the solution of the corresponding PE-fluid system

(1.10) \begin{equation}\left\{\begin{aligned} & \partial_t n + u \cdot \nabla n = \Delta n - \nabla \cdot \big(n S(x,n,c) \nabla c\big),& \qquad x\in\Omega,\,t>0,\\ \\[-12pt] & \qquad\quad u \cdot \nabla c = \Delta c-c+n ,& \qquad x\in\Omega,\,t>0,\\ \\[-12pt] & \partial_t u +(u\cdot\nabla)u + \nabla P = \Delta u +n \nabla \phi,& \qquad x\in\Omega,\,t>0,\\ \\[-12pt]& \qquad \quad \,\,\, \nabla \cdot u = 0, & \qquad x\in\Omega,\,t>0,\\ \\[-12pt] & \big(\nabla n-n S(x,n,c)\cdot\nabla c\big) \cdot \nu =\nabla c\cdot \nu=0, \quad u= 0,& \qquad x\in\partial\Omega,\,t>0,\\ \\[-12pt] &n(x,0)=n_0(x), \qquad u(x,0)= u_0(x), & \qquad x\in\Omega \end{aligned}\right.\end{equation}

in a setting as general as possible, where $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary.

Throughout this paper, we will suppose that the chemotactic sensitivity function $S=\big(S_{ij}\big)_{2\times 2}$ satisfies the requirements of regularity and the volume-filling hypothesis

(1.11) \begin{equation}S_{ij}(x,n_{\epsilon},c_{\epsilon})\in C^2\big(\overline{\Omega}\times[0,\infty)\times[0,\infty)\big) \qquad \mathrm{and} \qquad \big|S(x,n_{\epsilon},c_{\epsilon})\big| \le \frac{C_S}{ (1+n_{\epsilon})^{\alpha}}\end{equation}

for some constants $C_S>0$ and $\alpha>0$ , and that the initial data and the potential function fulfil

(1.12) \begin{equation}\left\{\begin{split}&n_0\in W^{2,\infty}(\Omega), \quad n_0\ge 0 \quad \textrm{and} \quad n_0\not\equiv 0\quad\textrm{in}\quad\overline{\Omega},\\ \\[-12pt] &c_0\in W^{1,\infty}(\Omega), \quad c_0\ge 0 \quad \textrm{and} \quad c_0\not\equiv 0\quad\textrm{in}\quad\overline{\Omega},\\ \\[-12pt] & u_0 \in W^{2,\infty}\big(\Omega; \, \mathbb R^2\big) \quad \textrm{with} \,\,\, \nabla \cdot u_0 \equiv0 \,\,\, \textrm{in} \,\,\, \Omega \,\,\, \textrm{ and} \,\,\, u_0=0 \,\,\, \textrm{on} \,\,\, \partial\Omega, \\ \\[-12pt] & \phi\in W^{2,\infty}(\Omega).\end{split}\right.\end{equation}

With the above framework, it was shown in [Reference Wang, Winkler and Xiang29] that for each fixed $\epsilon>0$ , system (1.9) admits a unique global bounded classical solution $(n_\epsilon,c_\epsilon,u_\epsilon,P_{\epsilon})$ satisfying $n_{\epsilon}\geq0$ and $c_{\epsilon}\geq0$ in $\Omega\times(0,\infty)$ .

Our aim is threefold: firstly, we show the global classical solutions $(n_\epsilon,c_\epsilon,u_\epsilon,P_{\epsilon})$ (not just a subsequence) of the full PP-fluid system (1.9) will converge to the solution (n, c, u, P) of the corresponding PE-fluid system (1.10) as $\epsilon\to 0$ . As a by-product, we obtain the global well-posedness of the PE-fluid system (1.10) for general large initial data. Secondly, we establish exponential time decay estimates of $(n_\epsilon,c_\epsilon,u_\epsilon)$ uniformly in $\epsilon$ for small initial cell mass, which in particular ensure an improvement of convergence rate with at most $\frac12$ -order growth on time t. Thirdly, we further investigate the convergence behaviour on $\epsilon$ and t through the numerical simulations of three different types of solution: the nontrivial and constant equilibriums, and the rotating aggregation (see Remark 5.1 for discussion).

Without loss of generality, we only need to focus on the case of $0<\alpha<\frac12$ . Under these assumptions, our main results are the following.

Theorem 1.1. Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with smooth boundary. Suppose that (1.11)–(1.12) hold for $\alpha>0$ , and that $(n_\epsilon,c_\epsilon,u_\epsilon,P_{\epsilon})$ solves the PP-fluid system (1.9) classically in $\Omega\times(0,\infty)$ . Then there exists a unique classical solution (n, c, u, P) to the PE-fluid system (1.10) in $\Omega\times(0,\infty)$ with the property that

\begin{equation*}\left\{\begin{split}& \|n_{\epsilon}(\cdot,t)-n(\cdot,t)\|_{L^2(\Omega)} + \|n_{\epsilon}(\cdot, s)- n(\cdot,s)\|_{L^2((0,t); W^{1, 2}(\Omega))} \le C_1e^{C_1t} \epsilon^{\frac12}, \\[3pt] & \qquad \qquad \qquad \|c_{\epsilon}(\cdot,s)- c(\cdot,s)\|_{L^2((0,t); W^{1, 2}(\Omega))} \le C_1e^{C_1t} \epsilon^{\frac12}, \\[3pt] & \|u_\epsilon(\cdot,t)-u(\cdot,t)\|_{L^{\infty}(\Omega)} + \| u_\epsilon(\cdot, s)- u(\cdot, s)\|_{L^2((0,t); W^{1, 2}(\Omega))} \le C_1e^{C_1t} \epsilon^{\frac12} \end{split} \right. \end{equation*}

for all $t\in(0, \infty)$ and some uniform positive constant $C_1$ . For each $\theta\in\big(\frac12,\frac34\big)$ and $p\ge 2$ , we also have

\begin{equation*}\left\{\begin{split}& \|{A}^\theta u_\epsilon(\cdot,t)-{A}^\theta u(\cdot,t)\|_{L^2(\Omega)} \le C_2e^{C_2t} \epsilon^{\frac12}, \\[3pt] & \|n_{\epsilon}(\cdot,t)-n(\cdot,t)\|_{L^p(\Omega)}\le C_3e^{C_3t} \epsilon^{\frac{1}{4}}\end{split} \right.\end{equation*}

for all $t\in(0, \infty)$ and some positive constants $C_2:=C_2(\theta)$ and $C_3:=C_3(p)$ .

Our second result further reveals that the above exponential growth in time t can be improved as at most $\frac12$ -order growth for small initial cell mass based on some new exponential time decay estimates of $(n_\epsilon,c_\epsilon,u_\epsilon)$ uniformly in $\epsilon$ . For simplicity, we will set $\overline{n}_0:=\frac{1}{|\Omega|}\int_{\Omega}n_0(x)dx$ .

Theorem 1.2. Under the assumptions of Theorem 1.1, there exists $\delta>0$ such that if

\begin{equation*}\|n_0\|_{L^{1}(\Omega)}\leq \delta,\end{equation*}

then the solutions $(n_{\epsilon}, c_{\epsilon}, u_\epsilon)$ to the PP-fluid system (1.9) satisfy the exponential time decay estimates uniformly in $\epsilon$

\begin{equation*} \|n_{\epsilon}(\cdot,t)-\overline{n}_0\|_{L^\infty({\Omega})} + \|c_{\epsilon}(\cdot,t)-\overline{n}_0\|_{W^{1,\,p}({\Omega})} +\| u_\epsilon(\cdot,t)\|_{L^{\infty}({\Omega})} \le C_1e^{-\mu t}\end{equation*}

for any $p>1$ and all $t\in(0, \infty)$ with some positive constants $\mu$ and $C_1$ . Moreover, there exists some uniform positive constant $C_2$ with the property that

\begin{equation*}\left\{\begin{split}& \|n_{\epsilon}(\cdot,t)-n(\cdot,t)\|_{L^2(\Omega)} + \|n_{\epsilon}(\cdot, s)- n(\cdot,s)\|_{L^2((0,t); W^{1, 2}(\Omega))} \le C_2 (1+t)^{\frac12} \epsilon^{\frac12}, \\[3pt] & \qquad \qquad \qquad \|c_{\epsilon}(\cdot,s)- c(\cdot,s)\|_{L^2((0,t); W^{1, 2}(\Omega))} \le C_2 (1+t)^{\frac12} \epsilon^{\frac12}, \\[3pt] & \| u_\epsilon(\cdot,t) - u(\cdot,t)\|_{W^{1, 2}(\Omega)}+ \| u_\epsilon(\cdot, s)- u(\cdot, s)\|_{L^2((0,t); W^{1, 2}(\Omega))} \le C_2 (1+t)^{\frac12} \epsilon^{\frac12} \end{split} \right. \end{equation*}

for all $t\in(0, \infty)$ . Furthermore, for each $\theta\in\big(\frac12, \frac34\big)$ and $p>2$ , we have

\begin{equation*} \|{A}^\theta u_\epsilon(\cdot,t)-{A}^\theta u(\cdot,t)\|_{L^2(\Omega)} \le C_3 (1+t)^{\frac34}\epsilon^{\frac12},\end{equation*}

and

\begin{equation*}\|n_{\epsilon}(\cdot,t)-n(\cdot,t)\|_{L^p(\Omega)}\le C_4 (1+t)^{\frac12} \epsilon^{\frac{2}{p^2}}\end{equation*}

for all $t\in(0, \infty)$ with some positive constants $C_3:=C_3(\theta)$ and $C_4:=C_4(p)$ .

Remark 1.1. In the current two-dimensional setting, Theorem 1.2 also improved the decay estimates obtained by [Reference Yu, Wang and Zheng40] in the sense that we removed the smallness restriction on $\|\nabla c_0\|_{L^2(\Omega)}$ and $\|u_0\|_{L^2(\Omega)}$ .

Key steps in our analysis. In Section 3, we concentrate upon the global existence of classical solution (n, c, u, P) to the PE-fluid system (1.10) as a limit of some subsequence of solutions $(n_\epsilon, c_\epsilon, u_\epsilon, P_{\epsilon})$ to the PP-fluid system (1.9). Thus, we need to derive some $\epsilon$ -independent estimates for $(n_\epsilon, c_\epsilon, u_\epsilon, P_{\epsilon})$ .

Unlike the PP-Stokes system studied in [Reference Li and Xiang19], the current mass conservation property $\|n_{\epsilon}(\cdot, t)\|_{L^1(\Omega)} = \|n_0\|_{L^1(\Omega)} $ and the regularity of $\|c_{\epsilon}(\cdot, t)\|_{L^1(\Omega)}$ (Lemma 2.1) cannot immediately provide the bounds for $u_\epsilon$ due to the convective term in equation (1.9) $_3$ . Instead, we will first analyse a combinational functional of the form

\begin{equation*}-\frac{1}{2\alpha}\|n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{2} + K \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2\end{equation*}

for some $K>0$ to gain the uniform $L^{2}$ space-time bounds for $\nabla n_{\epsilon}^\alpha$ and $\nabla c_{\epsilon}$ with respect to $\epsilon$ (Lemma 3.1), which ensures the $L^{2}$ spatial bound for $u_\epsilon$ and the $L^{2}$ space-time bound for $\nabla u_\epsilon$ (Lemma 3.2). We next improve our knowledge on the space-time $L^p$ uniform bound for $c_{\epsilon}$ for any $p\geq2$ (Lemma 3.3). Based on the above conclusions, we shall further establish the key $L^2$ boundedness of $\nabla u_\epsilon$ by an entropy-like estimate involving the combinational functional of the form

\begin{equation*}\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,}n_{\epsilon}+K\epsilon\int_{\Omega}|\nabla c_\epsilon|^2+M\int_{\Omega}|\nabla u_\epsilon|^2\end{equation*}

for some positive constants K and M (Lemma 3.4), which guarantees the time-independent spatial $L^p$ uniform bound for $u_\epsilon$ for any $p>1$ with respect to $\epsilon$ (Lemma 3.5). Thereafter, we will track the time evolution of the combinational functional

\begin{equation*}\|n_{\epsilon}(\cdot,t)\|_{L^{s}(\Omega)}^{s} + \epsilon \|\nabla c_{\epsilon}(\cdot,t)\|_{L^2(\Omega)}^2 \end{equation*}

for some s in Lemma 3.6. Then following from an induction argument (Corollary 3.1), we reach the $L^4$ regularity of $n_{\epsilon}$ (Corollary 3.2), which together with the damping effect of $c_{\epsilon}$ provides the uniform $L^2$ bound (Lemma 3.7) and the eventual $L^q$ bound for $ \nabla c_{\epsilon}$ (Lemma 3.8). These bounds imply the convergence of some subsequence of $(n_\epsilon, c_\epsilon, u_\epsilon, P_{\epsilon})$ (Lemma 3.9).

In Section 4, we shall first derive a linear growth estimate

\begin{equation*}\int_0^t \int_{\Omega}\partial_tc_{\epsilon} c \le C(1+t)\end{equation*}

for the mixed components $c_{\epsilon}$ and c using some subtle difference quotient estimates and the maximal regularity for parabolic equations and Stokes equation (Lemma 4.1). Then the basic energy methods and the variation-of-constants representation provide the convergence rate for general large initial data (Lemma 4.2-Lemma 4.5). Then we show that the solution $(n_{\epsilon}, c_{\epsilon}, u_\epsilon)$ of the PP-fluid system (1.9) exponentially decays to the constant steady state $(\overline{n}_0, \overline{n}_0, 0)$ uniformly in $\epsilon$ for appropriate small initial cell mass with $\overline{n}_0:=\frac{1}{|\Omega|}\int_{\Omega}n_0(x)dx$ (Lemmas 4.6, 4.7, Corollary 4.1, Lemma 4.8), which ensures that we can improve the growth in time t as at most $\frac12$ -order by investigating the time evolution of the mixed functional

\begin{equation*}K\|n_{\epsilon}(\cdot,t)-n(\cdot,t)\|_{L^2(\Omega)}^2 + \epsilon\|c_{\epsilon}(\cdot,t)\|_{L^{2}(\Omega)}^{2}+\|u_\epsilon(\cdot,t)-u(\cdot,t)\|_{L^2(\Omega)}^2\end{equation*}

for some $K>0$ (Lemma 4.9). The standard smoothing effect of Stokes semigroup and some energy estimates also entail the higher convergence of $u_\epsilon$ (Corollary 4.2 and Lemma 4.10) and $n_\epsilon$ (Lemma 4.11).

Notation: In the rest of this paper, we will suppose that $(n_{\epsilon},c_{\epsilon},u_\epsilon,P_{\epsilon})$ is a classical solution to the PP-fluid system (1.9) in $\Omega\times(0,\infty)$ with $\epsilon \in (0,1)$ . The positive constants $C, C_1, C_2, \cdots$ are independent of $\epsilon$ and t.

2. Preliminaries

In this section, we collected a few preliminaries. We begin with the mass conservation of cell density.

Lemma 2.1. Suppose that (1.11)–(1.12) hold. Then for all $\epsilon\in(0, 1)$ ,

(2.1) \begin{equation}\|n_\epsilon(\cdot,t)\|_{L^1(\Omega)}=\|n_0\|_{L^1(\Omega)}\qquad\mathrm{for\,\, all} \quad t\in (0,\infty),\end{equation}

and

(2.2) \begin{equation}\|c_\epsilon(\cdot,t)\|_{L^1(\Omega)} \leq \max\left\{\|n_0\|_{L^1(\Omega)}, \,\|c_0\|_{L^1(\Omega)}\right\}\qquad\mathrm{for\,\, all} \quad t\in (0,\infty).\end{equation}

Proof. The mass conversation (2.1) of $n_\epsilon$ can be obtained by taking an integration of equation (1.9) $_1$ over $\Omega$ . Similarly, integrating equation (1.9) $_2$ over $\Omega$ and using a comparison argument, we can obtain the $L^1$ boundedness (2.2) of $c_\epsilon$ .

Lemma 2.2. Suppose that (1.11)–(1.12) hold and that $\|n_{\epsilon}(\cdot,t)\|_{L^{s}(\Omega)}\le K$ , $(t\in(0, \infty))$ , for some $s>1$ and $K>0$ . Then there exists a positive constant C depending only on s, K and $c_0$ such that for all $\epsilon\in(0,1)$ ,

\begin{equation*}\|c_\epsilon(\cdot,t)\|_{L^{s}(\Omega)}\leq C \qquad \mathrm{for\,\, all} \quad t\in (0,\infty).\end{equation*}

Proof. Multiplying equation (1.9) $_2$ by $c_{\epsilon}^{s-1}$ and integrating by parts over $\Omega$ , we have

\begin{equation*} \frac{\epsilon}{s} \frac{d}{dt} \int_{\Omega}c_{\epsilon}^s +(s-1) \int_{\Omega}c_{\epsilon}^{s-2}|\nabla c_{\epsilon}|^2 + \int_{\Omega}c_{\epsilon}^s = \int_{\Omega}n_{\epsilon}c_{\epsilon}^{s-1} \le \frac{s-1}{s}\int_{\Omega}c_{\epsilon}^s+\frac{1}{s}\int_{\Omega}n_{\epsilon}^s\end{equation*}

and thus

\begin{equation*} \epsilon \frac{d}{dt} \int_{\Omega}c_{\epsilon}^s + s(s-1) \int_{\Omega}c_{\epsilon}^{s-2}|\nabla c_{\epsilon}|^2 + \int_{\Omega}c_{\epsilon}^s \le \int_{\Omega}n_{\epsilon}^s \le K^s\end{equation*}

for all $ t\in (0,\infty)$ . By a basic calculation, we deduce that

\begin{equation*}\int_{\Omega}c_{\epsilon}^s(\cdot,t) \le \max\left\{\int_{\Omega}c_0^s, \, K^s\right\} \qquad \mathrm{for\,\, all} \quad t\in (0,\infty).\end{equation*}

This completes the proof of Lemma 2.2.

Lemma 2.3. (Lemma 3.4 in [Reference Winkler38]) Let $a>0,\, T>0$ and $y\in C^0([0,T))\cap C^1(0,T)$ be such that

\begin{equation*}y'(t)+ay(t)\leq g(t) \qquad \mathrm{for\,\, all} \quad t\in (0,T),\end{equation*}

where the nonnegative function $g\in L_{loc}^1(\mathbb{R})$ has the property that $ \frac{1}{\tau}\int_{t}^{t+\tau}g(s)ds\leq b$ for all $t\in (0,T)$ with some $\tau>0$ and $b>0$ . Then

\begin{equation*}y(t)\leq y(0)+ \frac{b\tau}{1-e^{-a\tau}} \qquad \mathrm{for\,\, all} \quad t\in [0,T).\end{equation*}

3. Global existence of the PE-fluid system

In this section, we will establish the global existence of classical solution to the PE-fluid system (1.10) through a limit procedure in the PP-fluid system (1.9), which is highly nontrivial due to the loss of uniform parabolicity in $c_{\epsilon}$ equation. Our key idea is to obtain some necessary spatio-temporal estimates using a series of subtle coupled functional evolution estimates and bootstrap arguments.

3.1 The space-time $L^{2}$ bound for $\nabla u_\epsilon$ and $L^{p}$ bound for $c_\epsilon$

Lemma 3.1. Suppose that (1.11)–(1.12) hold. Then there exists some positive constant C such that for all $\epsilon\in(0,1)$ , we have

(3.1) \begin{equation}\int_t^{t+1}\int_{\Omega}|\nabla n_\epsilon^{\alpha}|^2 \le C\qquad \mathrm{and} \qquad\int_t^{t+1}\int_{\Omega}|\nabla c_\epsilon|^2 \le C \qquad \mathrm{for\,\, all} \quad t\geq0.\end{equation}

Proof. We first multiply equation (1.9) $_1$ by $n_{\epsilon}^{2\alpha-1}$ , integrate by parts over $\Omega$ and use the solenoidality of $u_\epsilon$ and the Young inequality to deduce that

(3.2) \begin{align} &\quad -\frac{1}{2\alpha}\frac{d}{dt}\int_{\Omega}n_{\epsilon}^{2\alpha}+(1-2\alpha)\int_{\Omega}n_{\epsilon}^{2\alpha-2}|\nabla n_{\epsilon}|^2 \nonumber\\[3pt] & =-\frac{1}{2\alpha}\int_{\Omega} u_\epsilon\cdot\nabla n_{\epsilon}^{2\alpha}+(1-2\alpha)\int_{\Omega}n_{\epsilon}^{2\alpha-1}\nabla n_{\epsilon}\cdot \big(S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\big) \nonumber\\[3pt] & =(1-2\alpha)\int_{\Omega}n_{\epsilon}^{2\alpha-1}\nabla n_{\epsilon}\cdot \big(S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\big) \nonumber\\[3pt] & \le \frac{1-2\alpha}{2} \int_{\Omega}n_{\epsilon}^{2\alpha-2} |\nabla n_{\epsilon}|^2+ \frac{1-2\alpha}{2}C_{S}^2\int_{\Omega}n_{\epsilon}^{2\alpha}(1+n_{\epsilon})^{-2\alpha}|\nabla c_{\epsilon}|^2 \nonumber\\[3pt] &\le \frac{1-2\alpha}{2} \int_{\Omega}n_{\epsilon}^{2\alpha-2} |\nabla n_{\epsilon}|^2+ C_1\int_{\Omega}|\nabla c_{\epsilon}|^2\end{align}

for all $t>0$ , where we also used the upper estimate (1.11) for S. To compensate the rightmost summand herein properly, we multiply equation (1.9) $_2$ by $c_{\epsilon}$ and utilise the solenoidality of $u_\epsilon$ again to find that

(3.3) \begin{equation}\frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}c_{\epsilon}^{2}+\int_{\Omega}|\nabla c_{\epsilon}|^2+\int_{\Omega}c_{\epsilon}^{2}= \int_{\Omega}n_{\epsilon}c_{\epsilon} \qquad \textrm{for all}\quad t>0.\end{equation}

For any fixed $\theta\in\left(1,\frac{1}{1-\alpha}\right)$ , we use the Hölder inequality to obtain

\begin{equation*}\int_{\Omega}n_{\epsilon}c_{\epsilon}\leq \|n_{\epsilon}\|_{L^{\theta}(\Omega)}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}=\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)} \qquad \textrm{for all}\quad t>0.\end{equation*}

Since the Sobolev embedding $W^{1,2}(\Omega)\hookrightarrow L^{\frac{\theta}{\theta-1}}(\Omega)$ and (2.2) imply that

\begin{equation*}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}^2\leq C_2\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^2+ C_2\|c_{\epsilon}\|_{L^{1}(\Omega)}^2\leq C_2\int_{\Omega}|\nabla c_{\epsilon}|^2+C_3 \qquad \textrm{for all}\quad t>0,\end{equation*}

we make use of the Young inequality to deduce that

(3.4) \begin{equation}\int_{\Omega}n_{\epsilon}c_{\epsilon}\leq \frac{1}{2C_2}\|c_{\epsilon}\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}^2+\frac{C_2}{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\le \frac12\int_{\Omega}|\nabla c_{\epsilon}|^2+\frac{C_3}{2C_2}+ \frac{C_2}{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\end{equation}

for all $t>0$ . In order to guarantee that the last summand here can be absorbed by the dissipated quantity in (3.2), we next apply the Gagliardo–Nirenberg inequality and the mass conservation (2.1) to see that

(3.5) \begin{equation}\frac{C_2}{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{\theta}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\leq C_4\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{\alpha\theta}}+C_4\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}\leq C_5\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+C_5\end{equation}

for all $t>0$ , whereupon substituting (3.4) and (3.5) into (3.3), we have

\begin{equation*}\epsilon\frac{d}{dt}\int_{\Omega}c_{\epsilon}^{2}+\int_{\Omega}|\nabla c_{\epsilon}|^2+2\int_{\Omega}c_{\epsilon}^{2}\leq 2C_5\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+C_6 \qquad \textrm{for all}\quad t>0\end{equation*}

with $C_6:=2C_5+\frac{C_3}{C_2}$ . This together with (3.2) entails that

\begin{align*}&\frac{d}{dt}\left\{\!-\frac{1}{2\alpha}\int_{\Omega}n_{\epsilon}^{2\alpha}+2C_1\epsilon\int_{\Omega}c_{\epsilon}^{2}\right\}+\frac{1-2\alpha}{2\alpha^2}\int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_1\int_{\Omega}|\nabla c_{\epsilon}|^{2}+4C_1\int_{\Omega}c_{\epsilon}^{2}\\[3pt] & \le 4C_1C_5\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+2C_1C_6\, \le \, \frac{1-2\alpha}{4\alpha^2}\int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_7 \qquad \textrm{for all}\quad t>0\end{align*}

and thus that

(3.6) \begin{equation}\frac{d}{dt}\left\{\!-\frac{1}{2\alpha}\int_{\Omega}n_{\epsilon}^{2\alpha}+2C_1\epsilon\int_{\Omega}c_{\epsilon}^{2}\right\}+\frac{1-2\alpha}{4\alpha^2}\int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_1\int_{\Omega}|\nabla c_{\epsilon}|^{2}+4C_1\int_{\Omega}c_{\epsilon}^{2} \le C_7\end{equation}

for all $t>0$ . Here we used the Young inequality in the last inequality of (3.6) due to $\frac{\theta-1}{\alpha\theta}\in(0,1)$ , which follows from $\theta\in\left(1,\frac1{1-\alpha}\right)$ . Then by setting

\begin{align*}y(t)&:=-\frac{1}{2\alpha}\|n_\epsilon^\alpha (\cdot,t)\|_{L^2(\Omega)}^2 + 2C_1\epsilon\|c_\epsilon (\cdot,t) \|_{L^2(\Omega)}^2,\\[3pt] g(t)&:=\frac{1-2\alpha}{4\alpha^2}\|\nabla n_\epsilon^\alpha (\cdot,t)\|_{L^2(\Omega)}^2 + C_1\|\nabla c_\epsilon (\cdot,t)\|_{L^2(\Omega)}^2,\end{align*}

and noticing that $4C_1\epsilon\int_{\Omega}c_{\epsilon}^2\le 4C_1\int_{\Omega}c_{\epsilon}^2 $ due to the fact $\epsilon\in(0,1)$ , we can conclude from (3.6) that

(3.7) \begin{equation}y'(t)+2y(t)+g(t)\leq C_{7}.\end{equation}

Since g(t) is nonnegative, we deduce from an ordinary differential inequality comparison argument that

(3.8) \begin{equation}y(t)\leq C_{8}:=\max\left\{\!-\frac{1}{2\alpha}\|n_0^\alpha\|_{L^2(\Omega)}^2+2C_1\|c_0\|_{L^2(\Omega)}^2, \frac{C_{7}}{2}\right\}\end{equation}

for all $t>0$ , which together with (3.7) yields that

\begin{equation*}\int_t^{t+1}g(s)ds\leq y(t)-y(t+1)-2\int_t^{t+1}y(s)ds+C_{7}\end{equation*}

for all $t\geq0$ . Due to $\alpha\in \left(0,\frac12\right)$ , we see from the Hölder inequality and the mass conservation (2.1) that

\begin{equation*}-y(t)\leq \frac{1}{2\alpha}\|n_\epsilon^\alpha (\cdot,t) \|_{L^2(\Omega)}^2 \le \frac{|\Omega|^{1-2\alpha}}{2\alpha}\|n_\epsilon (\cdot,t) \|_{L^1(\Omega)}^{2\alpha}=C_{9}:=\frac{|\Omega|^{1-2\alpha}}{2\alpha}\|n_0\|_{L^1(\Omega)}^{2\alpha},\end{equation*}

which together with (3.8) yields that

\begin{equation*}\frac{1-2\alpha}{4\alpha^2}\int_t^{t+1}\int_{\Omega}|\nabla n_\epsilon^{\alpha}|^2+C_1\int_t^{t+1}\int_{\Omega}|\nabla c_\epsilon|^2=\int_t^{t+1}g(s)ds\leq C_{10}:= C_{8}+3C_{9}+C_{7}\end{equation*}

for all $t\geq0$ , which entails (3.1).

Lemma 3.2. Suppose that (1.11)–(1.12) hold. Then there exists some positive constant C such that for all $\epsilon\in(0,1)$ we have

(3.9) \begin{equation}\| u_\epsilon (\cdot,t)\|_{L^{2}(\Omega)}\le C \qquad \mathrm{and} \qquad\int_t^{t+1}\int_{\Omega}|\nabla u_\epsilon|^2\le C \qquad \mathrm{for\,\, all} \quad t>0.\end{equation}

Proof. For any fixed $\theta\in\left(1,\frac1{1-\alpha}\right)$ , we test equation (1.9) $_3$ by $u_\epsilon$ and employ the Hölder inequality, the Sobolev embedding $W^{1,2}(\Omega)\hookrightarrow L^{\frac{\theta}{\theta-1}}(\Omega)$ , the Poincaré inequality and the Young inequality to obtain that

\begin{align*}\frac{1}{2} \frac{d}{dt} \|u_\epsilon\|_{L^2(\Omega)}^2 + \|\nabla u_\epsilon\|_{L^2(\Omega)}^2 &= \int_{\Omega}n_{\epsilon}\nabla\phi\cdot u_\epsilon \\[3pt] &\leq\|\nabla\phi\|_{L^\infty(\Omega)}\|n_{\epsilon}\|_{L^\theta(\Omega)}\|u_\epsilon\|_{L^{\frac{\theta}{\theta-1}}(\Omega)}\\[3pt] &\leq C_1 \|n_{\epsilon}\|_{L^\theta(\Omega)}\|\nabla u_\epsilon\|_{L^{2}(\Omega)}\\[3pt] &\leq \frac12\|\nabla u_\epsilon\|_{L^{2}(\Omega)}^2+\frac{C_1^2}{2}\|n_{\epsilon}\|_{L^\theta(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{align*}

By means of the Young inequality and (3.5), we have

(3.10) \begin{align}\frac{d}{dt} \|u_\epsilon\|_{L^2(\Omega)}^2+ \|\nabla u_\epsilon\|_{L^2(\Omega)}^2&\leq {C_1^2}\|n_{\epsilon}\|_{L^\theta(\Omega)}^2={C_1^2}\|n_{\epsilon}^\alpha\|_{L^\frac{\theta}{\alpha}(\Omega)}^\frac{2}{\alpha}\nonumber\\[3pt] &\leq C_{2}\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{\frac{2(\theta-1)}{\alpha\theta}}+C_{2}\leq \int_{\Omega}|\nabla n_{\epsilon}^\alpha|^2+C_{3} \qquad \mathrm{for\,\, all}\quad t>0\end{align}

due to $\frac{\theta-1}{\alpha\theta}\in(0,1)$ . Noticing that

\begin{equation*}\int_t^{t+1}\left(\int_{\Omega}|\nabla n_{\epsilon}^\alpha(\cdot,s)|^2+C_3\right)ds \le C_4\end{equation*}

due to Lemma 3.1, we then see from (3.10) and Lemma 2.3 that

\begin{equation*} \|u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2 \le C_5:= \|u_0\|_{L^2(\Omega)}^2+\frac{ C_4}{1-e^{-1}} \qquad \mathrm{for\,\, all}\quad t>0\end{equation*}

and thus that

\begin{equation*}\int_t^{t+1}\int_{\Omega}|\nabla u_\epsilon(\cdot,s)|^2ds\le \|u_0\|_{L^2(\Omega)}^2+ \int_t^{t+1}\left(\int_{\Omega}|\nabla n_{\epsilon}^\alpha(\cdot,s)|^2+C_3\right)ds \le C_6:= \|u_0\|_{L^2(\Omega)}^2+C_4\end{equation*}

for all $t\geq0$ . This completes the proof of Lemma 3.2.

We next intend to improve our knowledge on the space-time $L^p$ bound for $c_{\epsilon}$ for any $p\geq2$ . Indeed, for $2\leq p\leq3$ , the following lemma is a direct result of Lemma 3.1.

Lemma 3.3. Suppose that (1.11)–(1.12) hold. Then for each $p\geq2$ , we can find $C>0$ such that for all $\epsilon\in(0,1)$ ,

\begin{equation*}\int_t^{t+1}\int_{\Omega}c_{\epsilon}^p(\cdot,s)ds\le C \qquad \mathrm{for\,\, all}\quad t\geq0.\end{equation*}

Proof. Testing equation (1.9) $_2$ by $c_\epsilon^{p-1}$ , we obtain using the Hölder inequality that

\begin{equation*} \frac{\epsilon}{p}\frac{d}{dt}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2 + \frac{4(p-1)}{p^2}\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2= \int_{\Omega}n_{\epsilon}c_{\epsilon}^{p-1}\leq \|n_{\epsilon}\|_{L^r(\Omega)}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^{\frac{2(p-1)r}{p(r-1)}}(\Omega)}^{\frac{2(p-1)}{p}}\end{equation*}

for any $r>1$ and all $t>0$ . It then follows from the Gagliardo–Nirenberg inequality, the Young inequality and (2.2) that

\begin{align*}& \frac{\epsilon}{p}\frac{d}{dt}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2 + \frac{4(p-1)}{p^2}\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2\\[3pt]&\leq C_1\|n_{\epsilon}\|_{L^r(\Omega)}\left(\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^{\frac{2(pr-2r+1)}{pr}}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(r-1)}{pr}}+\|c_{\epsilon}^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(p-1)}{p}}\right)\\[3pt] &= C_1\|n_{\epsilon}\|_{L^r(\Omega)}\left(\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^{\frac{2(pr-2r+1)}{pr}}\|c_0\|_{L^{1}(\Omega)}^{\frac{r-1}{r}}+\|c_0\|_{L^{1}(\Omega)}^{p-1}\right)\\[3pt] &\leq C_2\|n_{\epsilon}\|_{L^r(\Omega)}\left(\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^{\frac{2(pr-2r+1)}{pr}}+1\right)\\[3pt] &\leq \frac{4(p-1)}{p^2}\|\nabla c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2+ C_3\|n_{\epsilon}\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}+C_3 \qquad \mathrm{for\,\, all}\quad t>0,\end{align*}

and thus that

(3.11) \begin{equation} \epsilon\frac{d}{dt}\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2 +p\|c_{\epsilon}^{\frac{p}{2}}\|_{L^2(\Omega)}^2\leq p C_3\|n_{\epsilon}\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}+pC_3 \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

Taking $r=\frac{p-2\alpha}{p-4\alpha}$ and applying the Gagliardo–Nirenberg inequality again, Lemma 3.1 and the mass conservation (2.1), we have

(3.12) \begin{align}\int_t^{t+1}\|n_{\epsilon}\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}&=\int_t^{t+1}\|n_{\epsilon}^\alpha\|_{L^{\frac{r}{\alpha}}(\Omega)}^{\frac{2r}{r-1}}\nonumber\\&\le C_4 \int_t^{t+1}\left(\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{2}\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{r-1}}+\|n_{\epsilon}^\alpha\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2r}{r-1}}\right)\nonumber\\&= C_4 \int_t^{t+1}\left(\|\nabla n_{\epsilon}^\alpha\|_{L^{2}(\Omega)}^{2}\|n_{\epsilon}\|_{L^{1}(\Omega)}^{\frac{2\alpha}{r-1}}+\|n_{\epsilon}\|_{L^{1}(\Omega)}^{\frac{2r\alpha}{r-1}}\right)\nonumber\\&\le C_5 \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

Consequently, setting

\begin{equation*}y(t):=\epsilon\|c_{\epsilon}^{\frac{p}{2}}(\cdot,t)\|_{L^2(\Omega)}^2, \qquad g(t)=p\|c_{\epsilon}^{\frac{p}{2}}(\cdot,t)\|_{L^2(\Omega)}^2, \qquad h(t):=p C_3\|n_{\epsilon}(\cdot,t)\|_{L^r(\Omega)}^{\frac{pr}{2r-1}}+pC_3,\end{equation*}

we can use (3.11) and the fact $p\|c_{\epsilon}^{\frac{p}{2}}(\cdot,t)\|_{L^2(\Omega)}^2> py(t)$ for any $\epsilon\in(0,1)$ to deduce that

\begin{equation*}y'(t)+py(t)\leq y'(t)+g(t)\leq h(t)\end{equation*}

for all $t>0$ and thus from Lemma 2.3 and (3.12) that

\begin{equation*}y(t)\leq y(0)+\frac{C_6}{1-e^{-p}}=\epsilon\|c_0^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\frac{C_6}{1-e^{-p}}<C_7:=\|c_0^{\frac{p}{2}}\|_{L^2(\Omega)}^2+\frac{C_6}{1-e^{-p}}\end{equation*}

for all $t>0$ with $C_6:=pC_3C_5+pC_3$ , which also implies that

\begin{equation*}\int_t^{t+1}g(s)ds\leq y(t)-y(t+1)+\int_t^{t+1}h(s)ds< C_7+C_6 \qquad \mathrm{for\,\, all}\quad t \ge 0.\end{equation*}

This completes the proof of Lemma 3.3.

3.2 A time-independent spatial $L^p$ bound for $u_\epsilon$

In this subsection, we derive further regularity of $u_\epsilon$ . Precisely, we will show the boundedness of $\|u_\epsilon(\cdot, t)\|_{L^p(\Omega)}$ for all $p\ge 1$ , which is based on the boundedness of $\|\nabla u_\epsilon(\cdot, t)\|_{L^2(\Omega)}$ obtained by a key entropy-like estimate.

Lemma 3.4. Suppose that (1.11)–(1.12) hold. Then there exists some positive constant C such that for all $\epsilon\in(0,1)$ we have

\begin{equation*} \|\nabla u_\epsilon(\cdot,t)\|_{L^2(\Omega)} \le C\qquad \mathrm{for\,\, all}\quad t>0.\end{equation*}

Proof. We will deduce our desired result by investigating the combinational functional of the form

\begin{equation*}\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,}n_{\epsilon}+K\epsilon\int_{\Omega}|\nabla c_\epsilon|^2+M\int_{\Omega}|\nabla u_\epsilon|^2\end{equation*}

with positive constants K and M to be determined.

For this purpose, since $\|A^{\frac12} u_\epsilon\|_{L^2(\Omega)}=\|\nabla u_\epsilon\|_{L^2(\Omega)}$ , we first apply the Helmholtz projection $\mathcal{P}$ to both sides of equation (1.9) $_3$ , multiply the result with $Au_\epsilon$ , integrate by parts over $\Omega$ and use the Hölder inequality, the $L^2$ boundedness of $\mathcal{P}$ , the Gagliardo–Nirenberg inequality and the Young inequality to obtain

\begin{align*} &\quad \frac{1}{2}\frac{d}{dt} \int_{\Omega} |\nabla u_\epsilon|^2 + \int_{\Omega} |Au_\epsilon|^2\\[3pt]& \le \|\mathcal{P}(u_\epsilon\cdot\nabla u_\epsilon)\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}+\|\mathcal{P}(n_\epsilon\nabla \phi)\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt]& \le C_1\| u_\epsilon\|_{L^4(\Omega)}\|\nabla u_\epsilon\|_{L^4(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}+C_1\|\nabla \phi\|_{L^\infty(\Omega)}\|n_\epsilon\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt] & \le C_2 \left(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\right)\left(\|A u_\epsilon\|_{L^2(\Omega)}^{\frac12}\|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\nabla u_\epsilon\|_{L^2(\Omega)}\right) \\[3pt] &\qquad \cdot \|A u_\epsilon\|_{L^2(\Omega)} + C_2 \|n_\epsilon\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt] & \le \|A u_\epsilon\|_{L^2(\Omega)}^2+C_3 \|\nabla u_\epsilon\|_{L^2(\Omega)}^{4} +C_3\|\nabla u_\epsilon\|_{L^2(\Omega)}^{2} +C_3 \|n_\epsilon\|_{L^2(\Omega)}^2\end{align*}

for all $t>0$ . Here we used (3.9) in the last inequality. Then, we have

(3.13) \begin{equation}\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2 \leq 2C_3 \|\nabla u_\epsilon\|_{L^2(\Omega)}^{4}+2C_3\|\nabla u_\epsilon\|_{L^2(\Omega)}^{2} +2C_3 \|n_\epsilon\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

We next estimate the last term on the right-hand side of (3.13). Due to $n_{\epsilon}>0$ in $\bar{\Omega}\times(0,\infty)$ , we may test (1.9) $_1$ by $\mathrm{ln\,}n_{\epsilon}+1$ to see from the integration by parts over $\Omega$ and the Young inequality, as well as the upper estimate (1.11) for S, that

\begin{align*}\frac{d}{dt}\int_{\Omega} n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}+\int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}&=\int_{\Omega}\nabla n_\epsilon\cdot\Big(S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_{\epsilon}\Big) \\[3pt] &\le \frac12 \int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}+\frac12 \int_{\Omega}n_{\epsilon}\Big|S(x,n_{\epsilon},c_{\epsilon})\Big|^2\cdot|\nabla c_{\epsilon} |^2\\[3pt] &\le \frac12 \int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}+\frac{C_S^2}{2}\int_{\Omega}n_{\epsilon} (1+n_{\epsilon})^{-2\alpha}|\nabla c_{\epsilon} |^2 \qquad \mathrm{for\,\, all}\quad t>0\end{align*}

and thus that

(3.14) \begin{equation}\frac{d}{dt}\int_{\Omega} n_{\epsilon}\mathrm{\,ln\,} n_{\epsilon}+\frac12\int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}\le \frac{C_S^2}{2}\int_{\Omega}n_{\epsilon}^{1-2\alpha}|\nabla c_{\epsilon} |^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

Noticing that

\begin{equation*}\int_{\Omega} n_{\epsilon}^2=\| \sqrt{n_\epsilon}\|_{L^4(\Omega)}^{4} \le C_4\|\nabla \sqrt{n_\epsilon}\|_{L^2(\Omega)}^{2}\| \sqrt{n_\epsilon}\|_{L^2(\Omega)}^{2} +C_4\| \sqrt{n_\epsilon}\|_{L^2(\Omega)}^{4}\le C_5 \int_{\Omega}\frac{|\nabla n_\epsilon|^2}{n_{\epsilon}}+C_5\end{equation*}

for all $t>0$ by the Gagliardo–Nirenberg inequality and the mass conservation (2.1), we deduce from the Young inequality that

\begin{equation*}\frac{C_S^2}{2}\int_{\Omega}n_{\epsilon} ^{1-2\alpha}|\nabla c_{\epsilon} |^2\leq \frac{1}{4C_5}\int_{\Omega} n_{\epsilon}^2 +C_6\int_{\Omega} |\nabla c_{\epsilon}|^{\frac{4}{1+2\alpha}} \qquad \mathrm{for\,\, all}\quad t>0,\end{equation*}

which together with (3.14) implies that

(3.15) \begin{equation}\frac{d}{dt}\int_{\Omega} n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}+\frac{1}{4C_5}\int_{\Omega}n_{\epsilon}^2 \le\frac12+ C_6\int_{\Omega} |\nabla c_{\epsilon}|^{\frac{4}{1+2\alpha}} \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

For the right-hand side of (3.15), we see from the Gagliardo–Nirenberg inequality and the Young inequality again that

\begin{align*}\frac12+ C_6\int_{\Omega} |\nabla c_{\epsilon}|^{\frac{4}{1+2\alpha}}&=\frac12+ C_6\|\nabla c_\epsilon\|_{L^{\frac{4}{1+2\alpha}}(\Omega)}^{\frac{4}{1+2\alpha}}\\[3pt] & \le \frac12+C_{7}\|\Delta c_\epsilon\|_{L^{2}(\Omega)}^{\frac{2}{1+2\alpha}}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{1+2\alpha}}+C_{7}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{4}{1+2\alpha}}\\[3pt] & \le \frac{1}{16C_5}\int_{\Omega}|\Delta c_{\epsilon}|^2+ C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8} \qquad \mathrm{for\,\, all}\quad t>0\end{align*}

due to $\alpha\in\left(0,\frac12\right)$ , and thus that

(3.16) \begin{equation}\frac{d}{dt}\int_{\Omega} n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}+\frac{1}{4C_5}\int_{\Omega}n_{\epsilon}^2 \le\frac{1}{16C_5}\int_{\Omega}|\Delta c_{\epsilon}|^2+ C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8} \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

To deal with the first integral on the right-hand side of (3.16), we test (1.9) $_2$ by $-\Delta c_{\epsilon}$ and integrate by parts over $\Omega$ to get

(3.17) \begin{align}\frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega} |\nabla c_{\epsilon}|^2+\int_{\Omega} |\Delta c_{\epsilon}|^2+\int_{\Omega} |\nabla c_{\epsilon}|^2=- \int_{\Omega}n_{\epsilon}\Delta c_{\epsilon}+ \int_{\Omega} (u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon} \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

For the first integral on the right-hand side of (3.17), it is clear that

(3.18) \begin{align}- \int_{\Omega}n_{\epsilon}\Delta c_{\epsilon}\leq\frac14\int_{\Omega} |\Delta c_{\epsilon}|^2+\int_{\Omega} n_{\epsilon}^2 \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

On the other hand, for the second integral on the right-hand side of (3.17), we apply the Hölder inequality to obtain

(3.19) \begin{align}\int_{\Omega} (u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon}\leq \|\Delta c_\epsilon\|_{L^2(\Omega)}\| {u_\epsilon}\|_{L^4(\Omega)}\|\nabla c_\epsilon\|_{L^4(\Omega)} \qquad \mathrm{for\,\, all}\quad t>0.\end{align}

Since

\begin{equation*} \int_{\Omega}|D^2 c_{\epsilon}|^2= \frac12 \int_{\partial\Omega} \nabla |\nabla c_{\epsilon} |^2 \cdot \nu - \int_{\Omega} \nabla \Delta c_{\epsilon} \cdot \nabla c_{\epsilon}= \frac12 \int_{\partial\Omega} \nabla |\nabla c_{\epsilon} |^2 \cdot \nu + \int_{\Omega} (\Delta c_{\epsilon})^2\end{equation*}

by the integration by parts and $\nabla c_{\epsilon}\cdot \nu=0$ , we can use the geometric property

(3.20) \begin{equation} \nabla |\nabla c_{\epsilon} |^2 \cdot \nu \le 2 C_{\Omega} |\nabla c_{\epsilon} |^2\end{equation}

with $C_{\Omega}$ an upper bound for the curvatures of $\partial\Omega$ (see Lemma 4.2 in [Reference Mizoguchi and Souplet22]), the trace theorem and the Gagliardo–Nirenberg inequality to see that

\begin{align*}\|D^2c_{\epsilon}\|_{L^2(\Omega)}^2& \le C_{\Omega}\|\nabla c_{\epsilon}\|_{L^{2}(\partial\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le C_{\Omega}\|\nabla c_{\epsilon}\|_{W^{\frac34, 2}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le C_9 \left( \|D^2c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac{11}6} \|c_{\epsilon}\|_{L^{1}(\Omega)}^{\frac16}+\|c_{\epsilon}\|_{L^{1}(\Omega)}^2 \right) + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le \frac12 \|D^2c_{\epsilon}\|_{L^{2}(\Omega)}^2 + \frac{C_{10}}2 \left( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \right) \qquad \text{for all}\quad t\in(0,\infty)\end{align*}

and thus that

(3.21) \begin{equation}\|D^2c_{\epsilon}\|_{L^2(\Omega)}^2 \le C_{10} \Big( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \Big) \qquad \text{for all}\quad t\in(0,\infty).\end{equation}

It then follows from the Gagliardo–Nirenberg inequality that

\begin{align*}\|\nabla c_\epsilon\|_{L^4(\Omega)}^2&\leq C_{11}\|D^2 c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{11}\|\nabla c_\epsilon\|_{L^2(\Omega)}^2\\&\le C_{11}C_{10}^{\frac12} \Big( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \Big) ^{\frac12}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{11}\|\nabla c_\epsilon\|_{L^2(\Omega)}^2 \\&\le C_{12}\|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{12}\|\nabla c_\epsilon\|_{L^2(\Omega)}+C_{12}\|\nabla c_\epsilon\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t>0,\end{align*}

which together with (3.19) by the Gagliardo–Nirenberg inequality, the Poincaré inequality and Lemma 3.2 entails that

(3.22) \begin{align} &\quad \int_{\Omega} (u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon} \nonumber\\[3pt]&\leq {C_{13}} \left(\|\Delta c_\epsilon\|_{L^2(\Omega)}^{\frac32}\|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12} + \|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)}\right)\| {u_\epsilon}\|_{L^4(\Omega)} \nonumber \\[3pt] & \le C_{14}\left(\|\Delta c_\epsilon\|_{L^2(\Omega)}^{\frac32}\|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\Delta c_\epsilon\|_{L^2(\Omega)} \|\nabla c_\epsilon\|_{L^2(\Omega)}^{\frac12} \right) \| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{\frac12}\| {u_\epsilon}\|_{L^2(\Omega)}^{\frac12} \nonumber \\[3pt] & \qquad +C_{14} \|\Delta c_\epsilon\|_{L^2(\Omega)}\|\nabla c_\epsilon\|_{L^2(\Omega)} \| \nabla{u_\epsilon}\|_{L^2(\Omega)} \nonumber \\[3pt] & \le {\frac14}\|\Delta c_\epsilon\|_{L^2(\Omega)}^{2}+C_{15}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{2}\| \nabla{c_\epsilon}\|_{L^2(\Omega)}^{2}+C_{15}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}\| \nabla{c_\epsilon}\|_{L^2(\Omega)} \nonumber \\[3pt] & \le {\frac14}\|\Delta c_\epsilon\|_{L^2(\Omega)}^{2}+\frac12\|\nabla c_\epsilon\|_{L^2(\Omega)}^{2}+C_{16}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{2}\| \nabla{c_\epsilon}\|_{L^2(\Omega)}^{2}+C_{16}\| \nabla{u_\epsilon}\|_{L^2(\Omega)}^{2}\end{align}

for all $t>0$ . Combining (3.18), (3.22) and (3.17), we can deduce that

\begin{align*}&\epsilon\frac{d}{dt}\int_{\Omega} |\nabla c_{\epsilon}|^2+\int_{\Omega} |\Delta c_{\epsilon}|^2+\int_{\Omega} |\nabla c_{\epsilon}|^2\leq 2\int_{\Omega} n_{\epsilon}^2\\[3pt] &\quad +2C_{16}\left(\int_{\Omega} |\nabla u_\epsilon|^2\right)\cdot\left(\int_{\Omega} |\nabla c_\epsilon|^2\right)+2C_{16}\int_{\Omega} |\nabla u_\epsilon|^2 \end{align*}

for all $t>0$ , which together with (3.16) and (3.13) yields that

\begin{align*}&\frac{d}{dt}\left( \int_{\Omega}n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon} +\frac{\epsilon}{16C_5}\int_{\Omega} |\nabla c_{\epsilon}|^2+\frac{1}{32C_3C_5}\int_{\Omega} |\nabla u_\epsilon|^2\right) + \frac{1}{16C_5}\int_{\Omega}n_{\epsilon}^2+\frac{1}{16C_5}\int_{\Omega} |\nabla c_{\epsilon}|^2\\[3pt] & \le \frac{C_{16}}{8C_5}\left(\int_{\Omega} |\nabla u_\epsilon|^2\right) \left(\int_{\Omega} |\nabla c_\epsilon|^2+1\right)+ \frac{1}{16C_5}\left(\int_{\Omega} |\nabla u_\epsilon|^2\right) \left(\int_{\Omega} |\nabla u_\epsilon|^2+1\right)\\ &\quad +C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}} + C_8 \\[3pt] & = \frac{1}{32C_3C_5}\int_{\Omega} |\nabla u_\epsilon|^2 \left(4C_3C_{16}\int_{\Omega} |\nabla c_{\epsilon}|^2 +2C_3\int_{\Omega} |\nabla u_\epsilon|^2+4C_3C_{16}+2C_3\right)\\ &\quad +C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8} \end{align*}

for all $t>0$ . Thus, by taking $K:=\frac{1}{16C_5}$ with $ M:=\frac{1}{32C_3C_5}$ and setting

\begin{equation*}y(t):=\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon} +K\epsilon\int_{\Omega} |\nabla c_{\epsilon}|^2+M\int_{\Omega} |\nabla u_\epsilon|^2,\qquad g(t):=K\int_{\Omega}n_{\epsilon}^2+K\int_{\Omega} |\nabla c_{\epsilon}|^2\end{equation*}

and

\begin{equation*}h(t):=4C_3C_{16}\int_{\Omega} |\nabla c_{\epsilon}|^2 +2C_3\int_{\Omega} |\nabla u_\epsilon|^2+4C_3C_{16}+2C_3,\qquad m(t):= C_{8}\| c_\epsilon\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8},\end{equation*}

we have

(3.23) \begin{equation}y'(t)+g(t)\leq h(t)\left\{y(t)+\frac{|\Omega|}{e}\right\}+m(t) \qquad \mathrm{for\,\, all}\quad t>0.\end{equation}

Here we used the fact that

(3.24) \begin{equation}-\int_{\Omega}n_{\epsilon} \mathrm{\,ln\,} n_{\epsilon}\leq\frac{|\Omega|}{e}, \qquad \mathrm{for\,\, all}\quad t>0\end{equation}

due to $\xi \mathrm{\,ln\,} \xi\geq-\frac{1}{e}$ for all $\xi>0$ .

According to the Gagliardo–Nirenberg inequality, the mass conservation (2.1) and Lemma 3.1, we can achieve that

\begin{align*}\int_{t-1}^{t}\|n_\epsilon(\cdot,s)\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}ds &=\int_{t-1}^{t}\|n_\epsilon^{\alpha}(\cdot,s)\|_{L^{\frac{1}{\alpha(1-\alpha)}}(\Omega)}^{\frac{2}{\alpha}}ds \\[3pt] &\le C_{17}\int_{t-1}^{t}\|\nabla n_\epsilon^{\alpha}(\cdot,s)\|_{L^{2}(\Omega)}^{2}\|n_\epsilon^{\alpha}(\cdot,s)\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2(1-\alpha)}{\alpha}}ds + C_{17}\int_{t-1}^{t}\|n_\epsilon^{\alpha}(\cdot,s)\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{2}{\alpha}}ds \\[3pt] & \le C_{18}\int_{t-1}^{t}\|\nabla n_\epsilon^{\alpha}(\cdot,s)\|_{L^{2}(\Omega)}^{2}ds +C_{18} \\[3pt] & \le C_{19} \qquad \mathrm{for\,\, all}\quad t\geq1.\end{align*}

This together with Lemmas 3.1 and 3.2 implies that

\begin{equation*}\int_{t-1}^{t}\left(\|n_\epsilon(\cdot,s)\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}+\|\nabla c_\epsilon(\cdot,s)\|_{L^{2}(\Omega)}^{2}+\|\nabla u_\epsilon(\cdot,s)\|_{L^{2}(\Omega)}^{2}\right)ds \le C_{20} \qquad \mathrm{for\,\, all}\quad t\geq1,\end{equation*}

and thus that for each fixed $t>0$ , we can find $t_\star\equiv t_\star(t;\epsilon)\geq0$ such that $t_\star\in\big( (t-1)_+, t\big)$ and

\begin{align*}&\|n_\epsilon(\cdot,t_\star)\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}+\|\nabla c_\epsilon(\cdot,t_\star)\|_{L^{2}(\Omega)}^{2}+\|\nabla u_\epsilon(\cdot,t_\star)\|_{L^{2}(\Omega)}^{2}\\[3pt] & \le C_{21}:=\max \left\{C_{20},\|n_0\|_{L^{\frac{1}{1-\alpha}}(\Omega)}^{2}+\|\nabla c_0\|_{L^{2}(\Omega)}^{2}+\|\nabla u_0\|_{L^{2}(\Omega)}^{2} \right\}.\end{align*}

By means of the elementary inequality $\xi \mathrm{\,ln\,} \xi\leq\frac{1-\alpha}{\alpha e}\xi^{\frac{1}{1-\alpha}}$ for all $\xi>0$ , we infer that

\begin{equation*}\int_{\Omega}n_{\epsilon}(\cdot,t_\star) \mathrm{\,ln\,} n_{\epsilon}(\cdot,t_\star) \leq \frac{1-\alpha}{\alpha e}\int_{\Omega}n_{\epsilon}^{\frac{1}{1-\alpha}}(\cdot,t_\star)\leq C_{22}:=\frac{1-\alpha}{\alpha e}C_{21}^{\frac{1}{2(1-\alpha)}}\end{equation*}

and thus that

\begin{equation*}y(t_\star)\leq C_{23}:=C_{22}+ KC_{21}+MC_{21}.\end{equation*}

On the other hand, we can see from Lemmas 3.1, 3.2 and 3.3 that

\begin{equation*}\int_{t-1}^{t}h(s)ds =\int_{t-1}^{t}\left(4C_3C_{16}\int_{\Omega} |\nabla c_{\epsilon}(\cdot,s)|^2+2C_3\int_{\Omega} |\nabla u_\epsilon(\cdot,s)|^2+4C_3C_{16}+2C_3\right)ds\leq C_{24}\end{equation*}

and

\begin{equation*}\int_{t-1}^{t}m(s)ds=\int_{t-1}^{t}\left(C_{8}\| c_\epsilon(\cdot,s)\|_{L^{\frac{1}{\alpha}}(\Omega)}^{\frac{1}{\alpha}}+C_{8}\right) ds \leq C_{25}\end{equation*}

for all $t\geq1$ . Thus integrating (3.23) from $t_\star$ to t, we can deduce that

\begin{align*}y(t)&\leq \Big(y(t_\star)+\frac{|\Omega|}{e}\Big) e^{\int_{t_\star}^{t}h(s)ds}+\int_{t_\star}^{t}e^{\int_{s}^{t}h(\sigma)d\sigma} m(s)ds \\[3pt] & \le \Big(C_{23}+\frac{|\Omega|}{e}\Big) e^{C_{24}}+ \int_{t_\star}^{t}e^{C_{24}} m(s)ds \le C_{26} \qquad \mathrm{for\,\, all}\quad t>0.\end{align*}

Whereupon, this together with (3.24) yields our desired conclusion.

Lemma 3.5. Suppose that (1.11)–(1.12) hold. Then for all $p>1$ , we can find some positive constant C such that for all $\epsilon\in(0,1)$ ,

\begin{equation*}\|u_\epsilon(\cdot,t)\|_{L^{p}(\Omega)}\leq C \qquad \mathrm{for\,\, all} \quad t\in (0,\infty).\end{equation*}

Proof. This is a direct consequence of Lemma 3.4, the Sobolev embedding $W^{1,2}(\Omega)\hookrightarrow L^{p}(\Omega)$ and the Poincaré inequality.

3.3 Time-independent spatial $\boldsymbol{L}^{\textbf{4}}$ bounds for $\boldsymbol{n}_{\boldsymbol\epsilon}$ and $\boldsymbol\nabla \boldsymbol{c}_{\boldsymbol\epsilon}$

We now improve our knowledge on the spatial regularity of $n_\epsilon$ by utilising a very subtle induction argument for $n_\epsilon$ , which together with the damping effect of $c_\epsilon$ will provide the key uniform $L^2$ bound for $\nabla c_\epsilon$ .

Lemma 3.6. Suppose that (1.11)–(1.12) hold. For any fixed $\widehat{r}\in\big(1, \frac{2}{2-\alpha}\big)$ and $p\ge 1$ , if it holds that for all $\epsilon\in(0,1)$ ,

(3.25) \begin{equation}\|n_{\epsilon}(\cdot,t)\|_{L^{p}(\Omega)} \leq K \qquad \mathrm{and} \qquad \|c_{\epsilon}(\cdot,t)\|_{L^{p}(\Omega)} \leq K\end{equation}

with some positive constant K, then for any

\begin{equation*}s \in \left(\max\left\{1, \, 2\alpha+\frac{p}{\widehat{r}}\right\}, \, \, 2\alpha + \frac{p(p+ 2\alpha \widehat{r} +2 - 2 \alpha)}{p+2\widehat{r}}\right),\end{equation*}

we have

\begin{equation*}\|n_{\epsilon}(\cdot,t)\|_{L^{s}(\Omega)} \leq C\qquad \mathrm{for\,\, all}\quad t\in (0,\infty)\end{equation*}

with some positive constant C.

Proof. Firstly, testing equation (1.9) $_1$ by $n_{\epsilon}^{s-1}$ , integrating by parts over $\Omega$ and making use of the Young inequality, the Hölder inequality and the upper estimate (1.11) for S, we have

\begin{align*} \quad \frac{1}{s}\frac{d}{dt} \int_{\Omega} n_{\epsilon}^{s}+(s-1) \int_{\Omega} n_{\epsilon}^{s-2}|\nabla n_{\epsilon}|^{2} &= (s-1) \int_{\Omega} n_{\epsilon}^{s-1} \nabla n_{\epsilon}\cdot \big(S(x,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big) \nonumber\\[3pt] &\le \frac{(s-1)}{4}\int_{\Omega} n_{\epsilon}^{s-2}|\nabla n_{\epsilon}|^{2}+(s-1) C_{{\mathcal S}}^2 \int_{\Omega} n_{\epsilon}^{s-2\alpha}|\nabla c_{\epsilon}|^2\end{align*}

and thus that

(3.26) \begin{align}\frac{d}{dt} \int_{\Omega}n_{\epsilon}^{s}+\frac{3(s-1)}{s} \int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2& \le s(s-1)C_{S}^2\int_{\Omega}n_{\epsilon}^{s-2\alpha}|\nabla c_{\epsilon}|^2 \nonumber \\& \le s(s-1)C_{S}^2 \left(\int_{\Omega}n_{\epsilon}^{\widehat{r}(s-2\alpha)}\right)^{\frac{1}{\widehat{r}}} \left(\int_{\Omega}|\nabla c_{\epsilon}|^{\frac{2\widehat{r}}{\widehat{r}-1}}\right)^{\frac{\widehat{r}-1}{\widehat{r}}}\end{align}

for all $ t\in (0,\infty)$ . For the first integral on the right-hand side of (3.26), we can deduce that

(3.27) \begin{align}\left(\int_{\Omega}n_{\epsilon}^{\widehat{r}(s-2\alpha)}\right)^{\frac{1}{\widehat{r}}} =\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2\widehat{r}(s-2\alpha)}{s}}(\Omega)}^{\frac{2(s-2\alpha)}{s}}& \le C_1\left(\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^{2}(\Omega)}^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}(s-2\alpha)}}\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2p}{s}}(\Omega)}^{\frac{p}{\widehat{r}(s-2\alpha)}} + \|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2p}{s}}(\Omega)}\right)^{\frac{2(s-2\alpha)}{s}} \nonumber \\& \le C_2 \left(\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 \right)^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}} + C_2\end{align}

for all $t\in(0,\infty)$ by (3.25) and $s>2\alpha+\frac{p}{\widehat{r}}$ , while for the second one, we have

(3.28) \begin{align}&\quad \left(\int_{\Omega}|\nabla c_{\epsilon}|^{\frac{2\widehat{r}}{\widehat{r}-1}}\right)^{\frac{\widehat{r}-1}{\widehat{r}}}=\|\nabla c_{\epsilon}\|_{L^{\frac{2\widehat{r}}{\widehat{r}-1}}(\Omega)}^2 \nonumber\\ & \le C_3 \left( \|D^2c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} \|c_{\epsilon}\|_{L^{p}(\Omega)}^{\frac{(\widehat{r}-1)p}{\widehat{r}(p+2)}}+\|c_{\epsilon}\|_{L^{p}(\Omega)} \right)^2 \nonumber \\& \le C_4 \left( \left( \|c_{\epsilon}\|_{L^{1}(\Omega)}^2 + \|\Delta c_{\epsilon}\|_{L^{2}(\Omega)}^2 \right)^{\frac{p+2\widehat{r}}{2\widehat{r}(p+2)}} \|c_{\epsilon}\|_{L^{p}(\Omega)}^{\frac{(\widehat{r}-1)p}{\widehat{r}(p+2)}} + \|c_{\epsilon}\|_{L^{p}(\Omega)} \right)^2 \nonumber \\& \le C_5 \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_6\end{align}

for all $t\in(0,\infty)$ by (3.21) and (3.25). Thus, by substituting (3.27) and (3.28) into (3.26), we can obtain

\begin{align*}& \frac{d}{dt}\int_{\Omega}n_{\epsilon}^{s}+\frac{3(s-1)}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 \\&\le C_7 \left(\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^{2}\right)^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}} \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_7 \left(\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2\right)^{\frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}}\\ &\quad + C_7 \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_7\end{align*}

for all $t\in(0,\infty)$ . Noticing that

\begin{equation*}0< \frac{\widehat{r}(s-2\alpha)-p}{\widehat{r}s}<1 \qquad \mathrm{and} \qquad 0< \frac{s(p+2\widehat{r})}{(p+2)(p+ 2\alpha \widehat{r})} <1\end{equation*}

due to $1< s< 2\alpha + \frac{p(p+ 2\alpha \widehat{r} +2 - 2 \alpha)}{p+2\widehat{r}}$ , we can use the Young inequality twice to obtain that

\begin{align*}& \frac{d}{dt}\int_{\Omega}n_{\epsilon}^{s} + \frac{3(s-1)}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 \\[3pt] & \le \frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 + C_{8} \left( \int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{s(p+2\widehat{r})}{(p+2)(2\alpha \widehat{r}+p)}} + C_{8} \left(\int_{\Omega}|\Delta c_{\epsilon}|^2\right)^{\frac{p+2\widehat{r}}{\widehat{r}(p+2)}} + C_{8} \\[3pt] & \le \frac{s-1}{s} \int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 + \int_{\Omega}|\Delta c_{\epsilon}|^2 +C_{9}\end{align*}

for all $t\in(0,\infty)$ and thus that

(3.29) \begin{equation}\frac{d}{dt}\int_{\Omega}n_{\epsilon}^{s}+\frac{2(s-1)}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2\le \int_{\Omega}|\Delta c_{\epsilon}|^2 + C_{9}\end{equation}

for all $t\in(0,\infty)$ .

On the other hand, in order to absorb the integral on the right-hand side of (3.29), we multiply equation (1.9) $_2$ by $-\Delta c_{\epsilon}$ and integrate on $\Omega$ to obtain that

(3.30) \begin{equation} \frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^2 + \int_{\Omega}|\Delta c_{\epsilon}|^2 +\int_{\Omega}|\nabla c_{\epsilon}|^2 = -\int_{\Omega}n_{\epsilon} \Delta c_{\epsilon} +\int_{\Omega}(u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon}\end{equation}

for all $t\in(0,\infty)$ . Since the interpolation and the mass conservation (2.1) imply that

\begin{align*}\int_{\Omega}n_{\epsilon}^2=\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{4}{s}}(\Omega)}^{\frac{4}{s}}& \le C_{10}\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^2(\Omega)}^{\frac{2}{s}} \|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{\frac{2}{s}}+C_{10}\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{\frac{4}{s}} \\[3pt] & \le C_{11}\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^2(\Omega)}^{\frac{2}{s}}+C_{11} \le \frac{s-1}{2s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2+C_{12} \qquad \text{for all}\quad t\in(0,\infty),\end{align*}

we have

(3.31) \begin{equation} - \int_{\Omega} n_{\epsilon} \Delta c_{\epsilon}\le \frac{1}{4}\int_{\Omega}|\Delta c_{\epsilon}|^2 + \int_{\Omega}n_{\epsilon}^2\le \frac{1}{4}\int_{\Omega}|\Delta c_{\epsilon}|^2 + \frac{s-1}{2s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2+C_{12}\end{equation}

for all $t\in(0,\infty)$ . Similarly, it follows from

\begin{equation*} \|\nabla c_{\epsilon}\|_{L^4(\Omega)} \le C_{13} \|\Delta c_{\epsilon}\|_{L^2(\Omega)}^{\frac56} \|c_{\epsilon}\|_{L^1(\Omega)}^{\frac16} + C_{13}\|c_{\epsilon}\|_{L^1(\Omega)} \le C_{14} \|\Delta c_{\epsilon}\|_{L^2(\Omega)}^{\frac56} + C_{14} \end{equation*}

and the boundedness of $\|u_\epsilon(\cdot, t)\|_{L^4(\Omega)}$ obtained in Lemma 3.5 that

(3.32) \begin{align} \int_{\Omega}(u_\epsilon\cdot\nabla c_{\epsilon})\Delta c_{\epsilon} & \le \|u_\epsilon\|_{L^4(\Omega)} \|\nabla c_{\epsilon}\|_{L^4(\Omega)} \|\Delta c_{\epsilon}\|_{L^2(\Omega)} \nonumber \\[3pt] & \le C_{14} \|u_\epsilon\|_{L^4(\Omega)} \left( \|\Delta c_{\epsilon}\|_{L^2(\Omega)}^{\frac{11}6} + \|\Delta c_{\epsilon}\|_{L^2(\Omega)} \right) \le \frac{1}{4}\int_{\Omega}|\Delta c_{\epsilon}|^2 + C_{15}\end{align}

for all $t\in(0,\infty)$ . Substituting (3.31) and (3.32) into (3.30), we can see

(3.33) \begin{equation} \epsilon\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^2+\int_{\Omega}|\Delta c_{\epsilon}|^2 +2\int_{\Omega}|\nabla c_{\epsilon}|^2\le \frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 + 2(C_{12}+ C_{15})\end{equation}

for all $t\in(0,\infty)$ .

Then combining (3.29) and (3.33), we obtain

\begin{equation*}\frac{d}{dt} \left\{ \int_{\Omega}n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2 \right\}+\frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2 +2\int_{\Omega}|\nabla c_{\epsilon}|^2\leq C_{16} \qquad \text{for all}\quad t\in(0, \infty)\end{equation*}

with $C_{16}:= C_{9} + 2(C_{12}+ C_{15})$ . To establish the uniform bound for the functional $\int_{\Omega}n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2$ , we apply the Gagliardo–Nirenberg inequality, the Young inequality and the mass conservation (2.1) to gain

\begin{equation*}\int_{\Omega}n_{\epsilon}^{s}=\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{2}(\Omega)}^{2}\le C_{17}\left(\|\nabla n_{\epsilon}^{\frac{s}{2}}\|_{L^{2}(\Omega)}^{\frac{2(s-1)}{s}}\|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{\frac2{s}}+ \|n_{\epsilon}^{\frac{s}{2}}\|_{L^{\frac{2}{s}}(\Omega)}^{2} \right)\le\frac{s-1}{s}\int_{\Omega}|\nabla n_{\epsilon}^{\frac{s}{2}}|^2+C_{18}\end{equation*}

for all $t\in(0,\infty)$ . Also noting $\epsilon\in(0,1)$ , we have

\begin{equation*}\frac{d}{dt}\left\{\int_{\Omega} n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2\right\}+ \left\{\int_{\Omega} n_{\epsilon}^{s} +\epsilon\int_{\Omega}|\nabla c_{\epsilon}|^2\right\}\leq C_{16}+C_{18}\end{equation*}

for all $t\in(0,\infty)$ . A direct calculation yields the desired result.

Corollary 3.1. Suppose that (1.11)–(1.12) hold. For any fixed $\widehat{r}\in\big(1, \frac{2}{2-\alpha}\big)$ , we define the sequence $\big\{ s_k\big\}_{k=0}^{\infty}$ by taking

\begin{equation*} s_0:=1, \quad s_k \in \left(\max\left\{1, \, 2\alpha+\frac{s_{k-1}}{\widehat{r}}\right\}, \, \, 2\alpha + \frac{s_{k-1}(s_{k-1}+ 2\alpha \widehat{r} +2 - 2 \alpha)}{s_{k-1}+2\widehat{r}}\right), \quad (k=1, 2, 3, \cdots).\end{equation*}

Then for every $k= 1, 2, 3, \cdots $ , there exists a positive constant $C_k$ such that for all $\epsilon\in(0,1)$ ,

\begin{equation*}\|n_{\epsilon}(\cdot,t)\|_{L^{s_k}(\Omega)} \leq C_k \qquad \mathrm{for\,\, all}\quad t\in (0,\infty).\end{equation*}

Proof. We can show the conclusion by an induction argument on k. Indeed, due to Lemmas 2.1 and 3.6, we see that the case $k=1$ is true. Then assuming that the conclusion is valid for some positive constant k, we can use Lemma 2.2 to obtain

\begin{equation*}\|c_\epsilon(\cdot,t)\|_{L^{s_k}(\Omega)}\le C_{1} \qquad \mathrm{for\,\, all} \quad t\in (0,\infty)\end{equation*}

with some $C_{1}>0$ , and thus have

\begin{equation*}\|n_{\epsilon}(\cdot,t)\|_{L^{s_{k+1}}(\Omega)} \le C_{2} \qquad \mathrm{for\,\, all}\quad t\in (0,\infty)\end{equation*}

with some positive constant $C_{2}$ by Lemma 3.6. This implies Corollary 3.1.

Corollary 3.2. Suppose that (1.11)–(1.12) hold. Then there exists $C>0$ such that for all $\epsilon\in(0,1)$ ,

\begin{equation*}\|n_{\epsilon}(\cdot,t)\|_{L^4(\Omega)} \leq C \qquad \mathrm{for\,\, all}\quad t\in (0,\infty).\end{equation*}

Proof. Let $\big\{ s_k\big\}_{k=0}^{\infty}$ be defined by Corollary 3.1. Then $ s_k> 2\alpha + \frac1{\widehat{r}} s_{k-1}$ , $(k=1, 2, 3, \cdots)$ , implies that

\begin{align*} & s_k > 2\alpha + \frac1{\widehat{r}} \left( 2\alpha + \frac1{\widehat{r}} s_{k-2}\right) > \cdots > 2\alpha \left(1 +\frac1{\widehat{r}} +\frac1{\widehat{r}^2} + \cdots + \frac1{\widehat{r}^{k-1}} \right)\\[3pt] &\quad + \frac1{\widehat{r}^k}s_0 = \frac{2\alpha\widehat{r}}{\widehat{r} -1 } + \frac1{\widehat{r}^k} \Big(s_0 - \frac{2\alpha\widehat{r}}{\widehat{r} -1} \Big)\end{align*}

for $k=1, 2, 3, \cdots$ . Noticing that $ \frac{2\alpha\widehat{r}}{\widehat{r} -1 }>4$ due to $\widehat{r}\in\big(1, \frac{2}{2-\alpha}\big)$ , we have $s_k\ge 4$ for k large enough. It then follows from Corollary 3.1 that $\|n_{\epsilon}(\cdot,t)\|_{L^{4}(\Omega)}$ is bounded in $(0, \infty)$ .

With the boundedness of $\|n_{\epsilon}(\cdot,t)\|_{L^{4}(\Omega)}$ at hand, we now turn back the proof of Lemma 3.6 to achieve the key boundedness of $\|\nabla c_{\epsilon}(\cdot,t)\|_{L^{2}(\Omega)}$ .

Lemma 3.7. Suppose that (1.11)–(1.12) hold. Then there exists $C>0$ such that for all $\epsilon\in(0,1)$ ,

\begin{align*}\\[-23pt] \|\nabla c_{\epsilon}(\cdot,t)\|_{L^2(\Omega)} \le C \qquad \mathrm{for\,\, all}\quad t\in (0,\infty).\end{align*}

Proof. By repeating the proof of (3.30), (3.31) and (3.32) in Lemma 3.6 and using Corollary 3.2, we can deduce that

\begin{equation*} \frac{\epsilon}{2}\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^2 + \int_{\Omega}|\Delta c_{\epsilon}|^2 +\int_{\Omega}|\nabla c_{\epsilon}|^2 \le \frac{1}{2}\int_{\Omega}|\Delta c_{\epsilon}|^2 + \int_{\Omega}n_{\epsilon}^2 + C_1 \le \frac{1}{2}\int_{\Omega}|\Delta c_{\epsilon}|^2 + C_2\end{equation*}

for all $t\in(0,\infty)$ . Then a direct calculation can complete the proof of Lemma 3.7.

Lemma 3.8. Suppose that (1.11)–(1.12) hold. Then for any $q>1$ , there exists $C>0$ such that for all $\epsilon\in(0,1)$ ,

\begin{equation*}\|\nabla c_{\epsilon}(\cdot,t)\|_{L^q(\Omega)}\le C \qquad \mathrm{for\,\, all}\quad t\in (0,\infty).\end{equation*}

Proof. We apply $\nabla$ to equation (1.9) $_2$ and test the resulting equation by $|\nabla c_{\epsilon}|^{2(p-1)}\nabla c_{\epsilon}$ with $p>\frac32$ to get

\begin{align*}& \frac{\epsilon}{2p}\frac{d}{dt}\int_\Omega |\nabla c_{\epsilon}|^{2p} - \int_{\Omega}|\nabla c_{\epsilon}|^{2(p-1)}\nabla c_{\epsilon}\cdot\Delta\nabla c_{\epsilon} + \int_{\Omega}|\nabla c_{\epsilon}|^{2p} = \int_{\Omega}|\nabla c_{\epsilon}|^{2(p-1)}\nabla c_{\epsilon}\cdot\nabla n_{\epsilon}\\ &\quad - \int_\Omega |\nabla c_{\epsilon}|^{2(p-1)} \nabla c_{\epsilon}\cdot\nabla\big(u_\epsilon\cdot\nabla c_{\epsilon}\big),\end{align*}

which together with the pointwise identity $2\nabla c_{\epsilon}\cdot\nabla\Delta c_{\epsilon}=\Delta|\nabla c_{\epsilon}|^2-2|D^2 c_{\epsilon}|^2$ and the integration by parts yields that

(3.34) \begin{align} &\frac{\epsilon}{2p} \frac{d}{dt}\int_\Omega |\nabla c_{\epsilon}|^{2p} + \frac{p-1}2 \int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} \left|\nabla|\nabla c_{\epsilon}|^2\right|^2 + \int_\Omega |\nabla c_{\epsilon}|^{2{(p-1)}}|D^2 c_{\epsilon}|^2 + \int_\Omega |\nabla c_{\epsilon}|^{2p} \nonumber \\[3pt] &= \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)} \nabla n_{\epsilon}\cdot\nabla c_{\epsilon} + (p-1) \int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2(p-2)} \nabla c_{\epsilon}\cdot\nabla|\nabla c_{\epsilon}|^2\nonumber\\ &\quad + \int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2{(p-1)}}\Delta c_{\epsilon} +\frac12 \int_{\partial\Omega}|\nabla c_{\epsilon}|^{2{(p-1)}}\frac{\partial|\nabla c_{\epsilon}|^2}{\partial\nu}\end{align}

for all $ t\in(0,\infty)$ . For the first term on the right-hand side of (3.34), it follows from the pointwise inequality $|\Delta c_{\epsilon}|^2\le 2|D^2 c_{\epsilon}|^2$ , the integration by parts and the Young inequality that

(3.35) \begin{align}& \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)} \nabla n_{\epsilon}\cdot\nabla c_{\epsilon} \nonumber \\[3pt] &=- \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)} n_{\epsilon}\Delta c_{\epsilon} - (p-1) \int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} n_{\epsilon}\nabla c_{\epsilon}\cdot\nabla|\nabla c_{\epsilon}|^2 \nonumber \\[3pt] &\le \sqrt{2} \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)}n_{\epsilon}|D^2 c_{\epsilon}| + (p-1)\int_\Omega n_{\epsilon}|\nabla c_{\epsilon}|^{2p-3}\big|\nabla|\nabla c_\epsilon|^2\big| \nonumber \\[3pt] &\le \frac12 \int_\Omega|\nabla c_{\epsilon}|^{2(p-1)}|D^2c_{\epsilon}|^2+\frac{p-1}4\int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} \left|\nabla|\nabla c_{\epsilon}|^2\right|^2 + p\int_\Omega n_{\epsilon}^2|\nabla c_{\epsilon}|^{2(p-1)},\end{align}

while for the second and third terms, we have

(3.36) \begin{align}& (p-1) \int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2(p-2)} \nabla c_{\epsilon}\cdot\nabla|\nabla c_{\epsilon}|^2 \nonumber \\[3pt] & \le\frac{p-1}8 \int_\Omega |\nabla c_{\epsilon}|^{2(p-2)} \left|\nabla|\nabla c_{\epsilon}|^2\right|^2 + 2(p-1)\int_\Omega| u_{\epsilon}|^2|\nabla c_{\epsilon}|^{2p}\end{align}

and

(3.37) \begin{align} &\int_\Omega(u_{\epsilon}\cdot\nabla c_{\epsilon})|\nabla c_{\epsilon}|^{2(p-1)}\Delta c_{\epsilon}\le \sqrt{2} \int_\Omega|u_{\epsilon}||\nabla c_{\epsilon}|^{2p-1}|D^2c_{\epsilon}|\le \frac12\int_\Omega|\nabla c_{\epsilon}|^{2(p-1)}|D^2 c_{\epsilon}|^2 \nonumber\\[3pt] &\quad + \int_\Omega|u_{\epsilon}|^2|\nabla c_{\epsilon}|^{2p}.\end{align}

For the last term on the right-hand side of (3.34), we know from the geometry property (3.20), the trace theorem, the Gagliardo–Nirenberg inequality, the Young inequality and Lemma 3.7 that

(3.38) \begin{align} \frac12\int_{\partial\Omega}|\nabla c_{\epsilon}|^{2(p-1)}\frac{\partial|\nabla c_{\epsilon}|^2}{\partial\nu}& \le C_1\int_{\partial\Omega}|\nabla c_{\epsilon}|^{2p}=C_1\big\||\nabla c_{\epsilon}|^p \big\|^2_{L^{2}(\partial\Omega)} \nonumber \\[3pt] & \le C_2\big\||\nabla c_{\epsilon}|^p\big\|^2_{W^{\frac34,2}(\Omega)} \nonumber \\[3pt] & \le C_3 \big\|\nabla|\nabla c_{\epsilon}|^p \big\|^{\frac{4p-1}{2p}}_{L^{2}(\Omega)} \big\||\nabla c_{\epsilon}|^p \big\|^{\frac{1}{2p}}_{L^{\frac2{p}}(\Omega)} + C_3\big\||\nabla c_{\epsilon}|^p \big\|_{L^{\frac{2}p}(\Omega)}^2 \nonumber \\[3pt] & \le \frac{p-1}{4p^2} \int_\Omega\left|\nabla |\nabla c_{\epsilon}|^p \right|^2 + C_4\end{align}

for all $t\in(0,\infty)$ . Inserting (3.35)–(3.38) into (3.34), we deduce that

\begin{align*}&\epsilon\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^{2p} + \frac{p-1}{2p} \int_{\Omega}\big|\nabla |\nabla c_{\epsilon}|^p\big|^2 + 2p \int_\Omega |\nabla c_{\epsilon}|^{2p} \le C_5 \int_{\Omega}n_{\epsilon}^2|\nabla c_{\epsilon}|^{2(p-1)}\\ & \quad + C_5 \int_{\Omega} |u_{\epsilon}|^2 |\nabla c_{\epsilon}|^{2p} + C_5\end{align*}

for all $t\in(0,\infty)$ . Noticing that

\begin{align*}C_5 \int_{\Omega} n_{\epsilon}^2 |\nabla c_{\epsilon}|^{2 (p-1)} & \le C_5 \|n_\epsilon\|_{L^4(\Omega)}^2 \|\nabla c_{\epsilon} \|_{L^{4(p-1)}(\Omega)}^{2(p-1)} \\[3pt] & \le C_6\|\nabla c_{\epsilon} \|_{L^{4(p-1)}(\Omega)}^{2(p-1)} =C_6 \big\||\nabla c_{\epsilon}|^p \big\|_{L^{\frac{4(p-1)}{p}}(\Omega)}^{\frac{2(p-1)}{p}} \\[3pt] & \le C_7 \left( \big\|\nabla|\nabla c_{\epsilon}|^p\big\|^{\frac{2p-3}{p}}_{L^{2}(\Omega)} \big\||\nabla c_{\epsilon}|^p\big\|^{\frac1{p}}_{L^{\frac2{p}}(\Omega)} + \big\||\nabla c_{\epsilon}|^p\big\|^{\frac{2(p-1)}{p}}_{L^{\frac2{p}}(\Omega)} \right) \\[3pt] & \le \frac{p-1}{4p} \big\|\nabla|\nabla c_{\epsilon}|^p\big\|^2_{L^{2}(\Omega)} + C_{8}\end{align*}

by Corollary 3.2 and Lemma 3.7, and that

\begin{align*} C_5 \int_{\Omega}|u_\epsilon|^2 |\nabla c_{\epsilon}|^{2p} & \le C_5 \||u_\epsilon|^2\|_{L^2(\Omega)} \big\||\nabla c_{\epsilon}|^{2p} \big\|_{L^2(\Omega)} \\[3pt] & \le C_9 \big\||\nabla c_{\epsilon}|^{2p} \big\|_{L^2(\Omega)} \\[3pt] & \le C_{10} \left( \big\|\nabla|\nabla c_{\epsilon}|^p \big\|_{L^{2}(\Omega)}^{\frac{2p-1}{p}}\big\||\nabla c_{\epsilon}|^p \big\|_{L^{\frac2{p}}(\Omega)}^{\frac1{p}} + \big\||\nabla c_{\epsilon}|^{p}\big\|^2_{L^{\frac2{p}}(\Omega)} \right) \\[3pt] & \le \frac{p-1}{4p} \big\|\nabla|\nabla c_{\epsilon}|^p \big\|^2_{L^{2}(\Omega)} + C_{11}\end{align*}

by Lemmas 3.5 and 3.7 again, we have

\begin{equation*}\epsilon\frac{d}{dt}\int_{\Omega}|\nabla c_{\epsilon}|^{2p} + 2p \int_\Omega |\nabla c_{\epsilon}|^{2p} \le C_{12} \qquad \text{for all} \quad t\in(0,\infty).\end{equation*}

Thus, a direct calculation yields the boundedness of $\|\nabla c_{\epsilon}(\cdot,t)\|_{L^{2p}(\Omega)}$ for any $p>\frac32$ . Then in light of the Hölder inequality we can complete the proof of Lemma 3.8.

Lemma 3.9. Suppose that (1.11)–(1.12) hold. Then there exist a sequence $\big\{\epsilon_j\big\}_{j=1}^{\infty}$ and a unique global classical solution (n, c, u, P) to the PE-fluid system (1.10) with the properties that

\begin{align*} n_{\epsilon_j} &\rightarrow n \qquad \mathrm{in} \quad C^0\big(\bar{\Omega}\times[0,\infty)\big), \\[3pt] n_{\epsilon_j} &\rightharpoonup n \qquad \mathrm{in} \quad L^2_{loc}\big((0,\infty);W^{1,2}(\Omega)\big), \\[3pt] c_{\epsilon_j} &\rightarrow c \qquad \mathrm{in} \quad L_{loc}^\infty\big((0,\infty);C^0(\bar{\Omega})\big) \bigcap L_{loc}^2\big((0,\infty);W^{1,2}(\Omega)\big), \\[3pt] \nabla c_{\epsilon_j} & \stackrel{*} {\rightharpoonup} \nabla c \qquad \mathrm{in} \quad \bigcap _{\hat{q}>2} L_{loc}^\infty \big((0,\infty);W^{1,\hat{q}}(\Omega)\big)\bigcap L_{loc}^\infty(\Omega\times(0,\infty)\big) \quad \mathrm{and} \\[3pt] u_{\epsilon_j} & \rightarrow u \qquad \mathrm{in} \quad C^0\big(\bar{\Omega}\times[0,\infty);\mathbb{R}^2\big) \bigcap C_{loc}^{2,1}\big(\bar{\Omega}\times(0,\infty);\mathbb{R}^2\big)\end{align*}

as $j \to \infty$ .

Proof. We know from Lemmas 3.8 and 3.5 that

\begin{equation*}\sup\limits_{\epsilon\in(0,1)}\|\nabla c_{\epsilon} \|_{L^{\infty}\left((0, \infty); L^4(\Omega)\right)} \le C \qquad \mathrm{and} \qquad\sup\limits_{\epsilon\in(0,1)}\| u_\epsilon \|_{L^{\infty}\left((0, \infty); L^4(\Omega)\right)} \le C.\end{equation*}

Therefore, in light of taking $d=2$ , $T=\infty$ , $\lambda=\infty$ , $q=4$ and $r=4$ in Theorem 1.1 of [Reference Wang, Winkler and Xiang30], we can complete the proof of Lemma 3.9. Here the convexity of $\Omega$ can actually be removed as pointed out by Remark (i) in [Reference Wang, Winkler and Xiang30] and the uniqueness of solution $(n,c,u.P)$ to the PE-fluid system (1.10) can be derived from a standard energy method and a bootstrap argument, and thus we omit the details.

4 Convergence rate

In the section, we will first derive the convergence rate for general large initial data and in particular obtain the convergence of the whole sequence $(n_{\epsilon},c_{\epsilon},u_\epsilon)$ (not just a subsequence as in Lemma 3.9). Then we derive some new exponential time decay estimates of $(n_\epsilon, c_\epsilon, u_\epsilon)$ with suitable small initial cell mass uniformly in $\epsilon$ . As a by-product, we improve the growth in time t as at most $\frac12$ -order.

For simplicity, letting (n, c, u, P) be the classical solution of the PE-fluid system (1.10) obtained in Lemma 3.9 and setting

\begin{equation*}\widehat{n}:=n_{\epsilon}-n, \qquad \widehat{c}:=c_{\epsilon}-c, \qquad \widehat{u}:=u_\epsilon-u, \qquad \mathrm{and} \qquad \widehat{P}:=P_\epsilon-P,\end{equation*}

we see that $(\widehat{n},\widehat{c},\widehat{u})$ is a solution to the following system:

(4.1) \begin{equation}\left\{\begin{split}& \,\, \, \partial_t \widehat{n} = \Delta\widehat{n}-u_\epsilon\cdot\nabla\widehat{n}-\widehat{u}\cdot\nabla n-\nabla\cdot\Big(\widehat{n} S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_\epsilon & \\ &\qquad \qquad + n S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla \widehat{c} + n\big(S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c)\big)\cdot\nabla c\Big), & x\in\Omega,\,\, t>0,\\ & \epsilon\partial_t c_{\epsilon} = \Delta \widehat{c}-u_\epsilon\cdot\nabla \widehat{c}-\widehat{u}\cdot\nabla c-\widehat{c}+\widehat{n}, & x\in\Omega,\,\, t>0,\\& \partial_t \widehat{u} = \Delta \widehat{u}-(u_\epsilon\cdot\nabla)\widehat{u}-(\widehat{u}\cdot\nabla) u-\nabla\widehat{P}+\widehat{n}\nabla\phi, & x\in\Omega,\,\, t>0,\\& \quad \nabla\cdot \widehat{u} = 0, & x\in\Omega,\,\, t>0,\\ & \Big( \nabla\widehat{n}-\widehat{n} S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla c_\epsilon - n S(x,n_{\epsilon},c_{\epsilon})\cdot\nabla \widehat{c} & \\ & \quad \qquad - n\big(S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c)\big)\cdot\nabla c\Big) \cdot\nu=0, \quad \nabla \widehat{c}\cdot \nu=0, \quad \widehat{u}=0, &\quad x\in\partial\Omega,\,\, t>0,\\ & \widehat{n}(x,0)=0,\quad \widehat{u}(x,0)=0, & x \in \Omega. \qquad \quad\end{split}\right.\end{equation}

4.1 The general large initial data case

Lemma 4.1. Suppose that (1.11)–(1.12) hold. There exists a positive constant C such that for each $\epsilon\in(0,1)$ ,

\begin{equation*}\int_0^t \int_{\Omega}\partial_tc_{\epsilon} c\le C(1+t) \qquad \text{for all} \quad t\in (0,\infty).\end{equation*}

Proof. Since

\begin{equation*} \int_{\Omega}\partial_tc_{\epsilon} c = \frac{d}{dt}\int_{\Omega}c_{\epsilon} c - \int_{\Omega}c_{\epsilon} \partial_t c \le \frac{d}{dt}\int_{\Omega}c_{\epsilon} c + \frac12 \left( \int_{\Omega} c_{\epsilon}^2 + \int_{\Omega} (\partial_t c) ^2\right) \qquad \text{for all} \quad t\in(0,\infty),\end{equation*}

we can see from the boundedness of $c_{\epsilon}$ and c, which follows from Lemma 3.8 and the Sobolev embedding, that

(4.2) \begin{align}\int_0^t \int_{\Omega}\partial_tc_{\epsilon} c&\le \int_{\Omega}c_{\epsilon}(\cdot, t) c(\cdot, t) - \int_{\Omega}c_0(\cdot) c(\cdot, 0) + \frac12 \left( \int_0^t \int_{\Omega} c_{\epsilon}^2 + \int_0^t \int_{\Omega} (\partial_t c) ^2\right) \nonumber \\[3pt] &\le \|c_{\epsilon}\|_{L^{\infty}(\Omega\times(0, \infty))} \|c\|_{L^{\infty}(\Omega\times(0, \infty))} |\Omega| + \frac12 \left( \|c_{\epsilon}\|_{L^{\infty}(\Omega\times(0, \infty))}^2 |\Omega| t + \int_0^t \int_{\Omega} (\partial_t c) ^2 \right) \nonumber \\[3pt] &\le C_1 (1+t) + \frac12 \int_0^t \int_{\Omega} (\partial_t c) ^2 \qquad \text{for all} \quad t\in(0,\infty).\end{align}

To establish the $L^2$ space-time estimate of $\partial_t c$ , we first derive the uniform estimates for the difference quotient

\begin{equation*}c_h(x,t):= \frac{c(x, t+h)-c(x,t)}{h} \qquad \text{for all} \quad t\in(\tau, \infty)\end{equation*}

with $\tau\in(0, \infty)$ and $h\in(-\tau, \infty)$ . It is easy to show that for each $t\in(\tau, \infty)$ , $c_h(\cdot,t) \in C^2(\overline{\Omega})$ is a classical solution of the homogeneous Neumann boundary-value problem for

\begin{equation*}-\Delta c_h(\cdot,t) + c_h(\cdot,t) = -u_h(\cdot, t)\cdot \nabla c(\cdot, t+h) - u(\cdot,t)\cdot\nabla c_h(\cdot,t) + n_h(\cdot, t)\end{equation*}

in $\Omega$ , where

\begin{equation*} u_h(\cdot, t) := \frac{u(x,t+h)-u(x,t)}{h}\qquad \mathrm{and} \qquad n_h(\cdot, t) := \frac{n(x,t+h)-n(x,t)}{h}. \end{equation*}

Then testing the above equation by $c_h(\cdot, t)$ , we obtain from integrating by parts over $\Omega$ , the solenoidality of u and the Hölder inequality that

\begin{align*}& \|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^2+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^2 \\[3pt] & =-\int_{\Omega}c_h(\cdot, t) u_h(\cdot, t)\cdot \nabla c(\cdot, t+h) - \frac12\int_{\Omega}u(\cdot,t)\cdot\nabla c_h^2(\cdot,t) + \int_{\Omega}c_h(\cdot, t) n_h(\cdot, t) \\[3pt] & =-\int_{\Omega}c_h(\cdot, t) u_h(\cdot, t)\cdot \nabla c(\cdot, t+h) + \int_{\Omega} c_h(\cdot, t) n_h(\cdot, t) \\[3pt] & \le \|c_h(\cdot, t)\|_{L^{4}(\Omega)}\|u_h(\cdot, t)\|_{L^{2}(\Omega)}\|\nabla c(\cdot, t+h)\|_{L^{4}(\Omega)} + \|c_h(\cdot, t)\|_{L^{2}(\Omega)}\|n_h(\cdot, t)\|_{L^{2}(\Omega)}\end{align*}

for all $t\in(\tau,\infty)$ , which together with the uniform boundedness of $\|\nabla c(\cdot, t+h)\|_{L^{4}(\Omega)}$ entails that

\begin{align*}& \|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^2+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^2 \\[3pt] & \le C_2 \|c_h(\cdot, t)\|_{L^{4}(\Omega)}\|u_h(\cdot, t)\|_{L^{2}(\Omega)} + \|c_h(\cdot, t)\|_{L^{2}(\Omega)}\|n_h(\cdot, t)\|_{L^{2}(\Omega)} \\[3pt] & \le C_{3}\Big(\|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^{\frac{1}2}\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^{\frac{1}2}+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}\Big)\|u_h(\cdot, t)\|_{L^{2}(\Omega)} +\|c_h(\cdot, t)\|_{L^{2}(\Omega)}\| n_h(\cdot, t)\|_{L^{2}(\Omega)} \nonumber\\[3pt] &\le \frac12 \Big( \|\nabla c_h(\cdot, t)\|_{L^{2}(\Omega)}^2+\|c_h(\cdot, t)\|_{L^{2}(\Omega)}^2\Big) + C_{4}\Big(\|u_h(\cdot, t)\|_{L^{2}(\Omega)}^2 + \|n_h(\cdot, t)\|_{L^{2}(\Omega)}^2 \Big)\end{align*}

for all $t\in(\tau,\infty)$ , and thus that

(4.3) \begin{align} &\int_\tau^t \int_{\Omega} |\nabla c_h| ^2 + \int_\tau^t \int_{\Omega} c_h^2 \le 2C_{4} \left( \int_\tau^t \int_{\Omega} n_h^2 + \int_\tau^t \int_{\Omega} |u_h|^2 \right) \nonumber\\[3pt]&\quad \le C_5 \left( \int_0^{t+1} \int_{\Omega} (\partial_tn) ^2 + \int_0^{t+1} \int_{\Omega} |\partial_tu|^2 \right),\end{align}

where in the last inequality we used the temporal version of Theorem 3(i) in Section 5.8.2, [Reference Evans7].

For the first integral on the right-hand side of (4.3), we can use the maximal regularity of parabolic equations (see Theorem 2.3 in [Reference Giga and Sohr10]) and the trace theorem to obtain

(4.4) \begin{align} \|{\partial_t n}\|_{L^{2}((0,t+1); L^2(\Omega))}& \le C_6\Big(\|u\cdot\nabla n\|_{L^{2}((0,t+1); L^2(\Omega))} + \|\nabla\cdot (nS(x,n,c) \cdot \nabla c) \|_{L^{2}((0,t+1); L^2(\Omega))} \nonumber \\ & \qquad + \| nS(x,n,c) \cdot \nabla c \|_{L^{2}((0,t+1); W^{\frac12,2}(\partial\Omega))} + \|n_0\|_{W^{1,2}(\Omega)} \Big) \nonumber \\& \le C_7\Big(\!\|u\cdot\nabla n\|_{L^{2}((0,t+1); L^2(\Omega))} + \|nS(x,n,c) \cdot \nabla c\|_{L^{2}((0,t+1); W^{1,2}(\Omega))} {+} \|n_0\|_{W^{1,2}(\Omega)} \Big) \nonumber \\&\le C_8 \Big(\! \|{u}\|_{L^{\infty}(\Omega\times(0,\infty))} \|\nabla n\|_{L^2((0, t+1); L^{2}(\Omega))} {+} \|{\nabla n}\|_{L^2((0, t+1); L^{2}(\Omega))} \|{\nabla c}\|_{L^\infty(\Omega\times(0,\infty))} \nonumber\\& \qquad + \|n\|_{L^{\infty}(\Omega\times(0,\infty))} \|{\nabla c}\|_{L^2((0, t+1); W^{1,2}(\Omega))} + \|n_0\|_{W^{1,2}(\Omega)} \Big) \nonumber \\&\le C_{9}(1+t)^{\frac12} \qquad \text{for all}\quad t\in(0,\infty),\end{align}

where in the last inequality, we used the bounds for n, u and $\nabla c$ derived from the proofs of Lemmas 2.2, 2.3 and 5.3 in [Reference Wang, Winkler and Xiang30], respectively, and the growth estimates $\|\nabla n\|^2_{L^2((0, t+1);L^2(\Omega))}\leq C(1+t)$ and $\|D^2 c\|^2_{L^{2}((0, t+1);L^2(\Omega))}\leq C(1+t)$ obtained by following the proof of Lemma 3.4 in [Reference Wang, Winkler and Xiang30] and by a direct integral in equation (1.10) $_2$ , respectively. On the other hand, for the second integral on the right-hand side of (4.3), we can utilise the Sobolev maximal regularity of Stokes equation (see Theorem 2.1 in [Reference Giga and Sohr10]) and $\|n\|^2_{L^{2}((0, t+1);L^2(\Omega))}\leq C(1+t)$ to obtain that

\begin{align*}&\int_{0}^{t+1}\|\partial_t u(\cdot,s)\|_{L^{2}(\Omega)}^2ds+\int_{0}^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds\\&\qquad\leq C_{10} \int_{0}^{t+1}\Big(\big\|u\cdot\nabla u(\cdot,s)\big\|_{L^{2}(\Omega)}^2 + \big\|n\nabla\phi(\cdot,s)\big\|_{L^{2}(\Omega)}^2\Big)ds\\& \qquad\le C_{10}\|u\|_{L^{\infty}(\Omega\times(0, \infty))}^2\int_{0}^{t+1}\|\nabla u(\cdot,s)\|_{L^{2}(\Omega)}^2ds +C_{10}\|\nabla\phi\|_{L^{\infty}(\Omega)}^2\int_{0}^{t+1}\|n(\cdot,s)\|^2_{L^{2}(\Omega)}ds \\&\qquad \le C_{11}\int_{0}^{t+1}\Big(\| u(\cdot,s)\|_{H^{2}(\Omega)}\| u(\cdot,s)\|_{L^{2}(\Omega)}+\| u(\cdot,s)\|_{L^{2}(\Omega)}^2\Big)ds+C_{11}(1+t)\\& \qquad \le \frac12\int_{0}^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds +C_{12}\int_{0}^{t+1}\| u(\cdot,s)\|_{L^{2}(\Omega)}^2ds+C_{12}(1+t) \\&\qquad \le \frac12\int_{0}^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds + C_{13} (1+t) \qquad \mathrm{for\,\, all} \quad t\in (0,\infty)\end{align*}

and thus that

(4.5) \begin{equation} \int_0^{t+1}\|\partial_t u(\cdot,s)\|_{L^{2}(\Omega)}^2ds +\frac12 \int_0^{t+1}\| u(\cdot,s)\|_{H^{2}(\Omega)}^2ds \le C_{13}(1+t) \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation}

Consequently, inserting (4.4) and (4.5) into (4.3), we see that

\begin{equation*} \int_\tau^t \int_{\Omega} |\nabla c_h| ^2 + \int_\tau^t \int_{\Omega} c_h^2 \le C_{14}(1+t):=C_5\big( C_{9}^2(1+t) + C_{13} (1+t)\big) \qquad \mathrm{for\,\, all}\quad t\in(\tau, \infty) \end{equation*}

and that there exists $(h_i)_{i\in\mathbb N}$ satisfying $h_i\to 0$ and $c_{h_i} \to \partial_tc$ in $L^2(\Omega\times (\tau,t))$ as $i\to\infty$ with the same bound $\int_\tau^t \int_{\Omega} (\partial_tc)^2 \le C_{14}(1+t)$ . Since $C_{14}$ is independent of $\tau$ , we may take $\tau \searrow 0$ to obtain $\int_0^t \int_{\Omega} (\partial_tc)^2 \le C_{14}(1+t)$ , which together with (4.2) completes the proof of Lemma 4.1.

Lemma 4.2. Suppose that (1.11)–(1.12) hold. Then there exists $C>0$ such that for each $\epsilon\in(0,1)$ ,

\begin{equation*}\|\widehat{n}(\cdot,t)\|_{L^2(\Omega)}+\|\widehat{u}(\cdot,t)\|_{L^2(\Omega)} \le Ce^{Ct} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

and

\begin{equation*}\|\widehat{n}\|_{L^2((0,t); W^{1, 2}(\Omega))}+\|\widehat{c}\|_{L^2((0,t); W^{1, 2}(\Omega))} + \|\widehat{u}\|_{L^2((0,t); W^{1, 2}(\Omega))}\le Ce^{Ct} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. The proof is parallel to that of Lemma 5.2 in [Reference Li and Xiang19] and thus we omit the details here.

Lemma 4.3. Suppose that (1.11)–(1.12) hold. Then there exists $C>0$ such that for each $\epsilon\in(0,1)$ ,

\begin{equation*}\|\nabla\widehat{u}(\cdot,t)\|_{L^2(\Omega)} \le Ce^{Ct} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. Applying the Helmholtz projection $\mathcal{P}$ to both sides of equation (4.1) $_3$ and testing the resulting equation with $A\widehat{u}$ , we obtain

\begin{align*}&\frac{1}{2}\frac{d}{dt}\|\nabla \widehat{u}\|_{L^2(\Omega)}^2 + \|A\widehat{u}\|_{L^2(\Omega)}^2\\[3pt]&\quad =-\int_{\Omega} \mathcal{P}\big(u_\epsilon\cdot\nabla \widehat{u} + \widehat{u}\cdot\nabla u - \widehat{n}\nabla \phi\big) \cdot A\widehat{u} \\[3pt] & \quad \leq C_1 \Big( \| u_\epsilon\|_{L^4(\Omega)}\|\nabla\widehat{u}\|_{L^4(\Omega)} + \|\nabla u\|_{L^2(\Omega)} \| \widehat{u}\|_{L^\infty(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)} \| \nabla \phi\|_{L^\infty(\Omega)} \Big) \|A \widehat{u}\|_{L^2(\Omega)}\\[3pt] &\quad \le C_2\Big(\|A \widehat{u}\|_{L^2(\Omega)}^{\frac34}\| \widehat{u}\|_{L^2(\Omega)}^{\frac14} + \|A \widehat{u}\|_{L^2(\Omega)}^{\frac12}\| \widehat{u}\|_{L^2(\Omega)}^{\frac12}+\|\widehat{u}\|_{L^2(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)} \Big)\|A \widehat{u}\|_{L^2(\Omega)}\\[3pt] &\quad \le \|A \widehat{u}\|_{L^2(\Omega)}^2+C_3 \|\widehat{u}\|_{L^2(\Omega)}^{2} +C_3 \|\widehat{n}\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty),\end{align*}

where we used the uniform boundedness of $\| u_\epsilon\|_{L^4(\Omega)}$ and the boundedness of $\|\nabla u\|_{L^2(\Omega)}$ . It then follows from Lemma 4.2 and $\widehat{u}(\cdot,0)=0$ that

(4.6) \begin{equation}\|\nabla \widehat{u}(\cdot,t)\|_{L^2(\Omega)}^2 \leq 2C_3\int_0^{t}\Big( \|\widehat{n}(\cdot,s)\|_{L^2(\Omega)}^2 + \|\widehat{u}(\cdot,s)\|_{L^2(\Omega)}^{2}\Big)ds\le C_4 e^{C_4t}\epsilon\end{equation}

for all $t\in(0,\infty)$ . This completes the proof of Lemma 4.3.

Lemma 4.4. Suppose that (1.11)–(1.12) hold. Then for any given $\theta\in\big(\frac12,\frac34\big)$ , there exists $C(\theta)>0$ such that for each $\epsilon\in(0,1)$ ,

\begin{equation*}\|A^\theta\widehat{u}(\cdot,t)\|_{L^2(\Omega)} \le C(\theta)e^{C(\theta)t} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

In particular, there exists a positive constant C such that

\begin{equation*}\|\widehat{u}(\cdot,t)\|_{L^\infty(\Omega)} \le Ce^{Ct} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. Using the variation-of-constants representation of $\widehat{u}$ , we have

(4.7) \begin{align}A^{\theta}\widehat{u}(\cdot,t)&:= -\int^t_0{A}^{\theta} e^{-(t-s){A}}\mathcal{P}\big(u_\epsilon\cdot\nabla \widehat{u} + \widehat{u}\cdot\nabla u - \widehat{n}\nabla\phi \big)(\cdot,s) ds.\end{align}

Setting $ f(\cdot,s) := \mathcal{P}\big(u_\epsilon\cdot\nabla \widehat{u} + \widehat{u}\cdot\nabla u - \widehat{n}\nabla\phi \big)(\cdot,s)$ , we can use the Hölder inequality, the Gagliardo–Nirenberg inequality, the boundedness of $u_\epsilon$ , Lemmas 4.2 and 4.3 to obtain

\begin{align*}\|f(\cdot,s)\|_{L^2(\Omega)}&\leq \|u_\epsilon\|_{L^\infty(\Omega)}\|\nabla\widehat{u}\|_{L^2(\Omega)} + \|\widehat{u}\|_{L^4(\Omega)}\|\nabla{u}\|_{L^4(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)}\|\nabla\phi\|_{L^\infty(\Omega)}\\[3pt]&\le C_1 \|\nabla\widehat{u}\|_{L^2(\Omega)} + C_1\|\nabla{u}\|_{L^4(\Omega)}\left(\|\nabla\widehat{u}\|_{L^2(\Omega)}^{\frac12}\|\widehat{u}\|_{L^2(\Omega)}^{\frac12}+\|\widehat{u}\|_{L^2(\Omega)}\right)+C_1\|\widehat{n}\|_{L^2(\Omega)}\\[3pt] &\le C_2e^{C_2s}\epsilon^{\frac12}+ C_2\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}e^{C_2s}\epsilon^{\frac12}\end{align*}

and thus have

(4.8) \begin{align}\|{A}^{\theta}\widehat{u}(\cdot,t)\|_{L^2(\Omega)}&\le C_3 \int^t_0(t-s)^{-\theta}e^{-\lambda(t-s)}\| f(\cdot,s)\|_{L^2(\Omega)}ds \nonumber \\[3pt] &\le C_4\epsilon^{\frac12}\int^t_0(t-s)^{-\theta}e^{-\lambda(t-s)+C_2s}ds \nonumber\\[3pt] &\qquad +C_4\epsilon^{\frac12}\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}(t-s)^{-\theta}e^{-\lambda(t-s)+C_2s}ds \nonumber \\[3pt] &\le C_5e^{C_5t}\epsilon^{\frac12} + C_4\epsilon^{\frac12}\left(\int^t_0\big((t-s)^{-\theta}e^{-\lambda(t-s)+C_2s}\big)^\frac43 ds\right)^{\frac34}\left(\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds\right)^{\frac14} \nonumber \\[3pt] &\le C_5e^{C_5t}\epsilon^{\frac12} + C_6e^{C_6t}\epsilon^{\frac12}\left(\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds\right)^{\frac14}\end{align}

for some $\lambda>0$ due to $\widehat{u}(\cdot, 0)=0$ and $\theta\in\big(\frac12,\frac34\big)$ . For the last integral on the right-hand side of (4.8), we apply the Gagliardo–Nirenberg inequality, (4.5) and the boundedness of $\|{u}\|_{L^\infty(\Omega\times(0,\infty))}$ to obtain

(4.9) \begin{align}\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds&\le \int^t_0\Big(\|u(\cdot,s)\|_{H^2(\Omega)}^2\|u(\cdot,s)\|_{L^\infty(\Omega)}^2+\|u(\cdot,s)\|_{L^\infty(\Omega)}^4\Big) ds \nonumber\\[3pt]&\le \|{u}\|^2_{L^\infty(\Omega\times(0,\infty))}\int^t_0\|u(\cdot,s)\|_{H^2(\Omega)}^2ds+\|{u}\|^4_{L^\infty(\Omega\times(0,\infty))}|\Omega|t \nonumber\\[3pt]&\le C_7(1+t)\end{align}

for all $t\in(0,\infty)$ . This together with (4.8) and the fact $1+t\leq e^t$ yields that

\begin{equation*}\|{A}^{\theta}\widehat{u}(\cdot,t)\|_{L^2(\Omega)}\le C_8 e^{C_8t} \epsilon^{\frac12} \qquad \text{for all} \quad t\in(0, \infty).\end{equation*}

Meanwhile, the estimate of $\|\widehat{u}(\cdot,t)\|_{L^{\infty}(\Omega)}$ will follow from $D({A}^\theta)\hookrightarrow L^\infty(\Omega)$ due to $\theta\in\left(\frac12,\frac34\right)$ . This completes the proof of Lemma 4.4.

Lemma 4.5. Suppose that (1.11)–(1.12) hold. Then for any $p\ge 2$ , there exists $C(p)>0$ such that for each $\epsilon\in(0,1)$ ,

\begin{equation*} \|\widehat{n}(\cdot,t)\|_{L^p(\Omega)}\leq C(p)e^{C(p)t}\epsilon^{\frac{1}{4}} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. The proof is parallel to that of Lemma 5.4 in [Reference Li and Xiang19] and thus we omit the details here.

Proof of Theorem 1.1. We can conclude the desired result using Lemmas 3.9, 4.2, 4.4 and 4.5.

4.2 The small initial cell mass case

In this subsection, we first show the time decay of solutions to the PP-fluid system (1.9) with suitable small initial cell mass, which will be used to improve the exponential growth on time t in the fast signal diffusion limit procedure.

Lemma 4.6. Suppose that (1.11)–(1.12) hold. Then if

\begin{equation*}\|n_0\|_{L^1(\Omega)}\leq \delta\end{equation*}

holds for some suitable small $\delta>0$ , then there exist two positive constants C and $\mu$ such that

(4.10) \begin{equation} \|n_{\epsilon}(\cdot,t)-\overline{n}_0\|_{L^2({\Omega})}\leq Ce^{-\mu t} \qquad \mathrm{for \,\, all} \quad t\in(0,\infty)\end{equation}

provided that $\epsilon$ is suitable small.

Proof. Firstly, we multiply equation (1.9) $_1$ by $n_{\epsilon}-\overline{n}_0$ and integrate by parts over $\Omega$ to obtain

\begin{align*}\frac{1}{2}\frac{d}{dt}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + \|\nabla n_\epsilon\|_{L^2(\Omega)}^2&=\int_{\Omega}\nabla n_{\epsilon}\cdot \big(n_{\epsilon} S(x,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)\\[3pt] &\quad \le \frac12\|\nabla n_\epsilon\|_{L^2(\Omega)}^2+\frac{C_S^2}{2}\int_{\Omega}n_{\epsilon}^2|\nabla c_{\epsilon}|^2\end{align*}

for all $t\in(0,\infty)$ , and thus have

\begin{align*}\frac{d}{dt}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + \|\nabla n_\epsilon\|_{L^2(\Omega)}^2&\le C_S^2\int_{\Omega}n_{\epsilon}^2|\nabla c_{\epsilon}|^2 \\[3pt] &\le 2C_S^2\int_{\Omega}(n_\epsilon-\overline{n}_0)^2|\nabla c_{\epsilon}|^2+ 2C_S^2\int_{\Omega}\overline{n}_0^2|\nabla c_{\epsilon}|^2 \\[3pt] &\le 2C_S^2 \|n_\epsilon-\overline{n}_0\|_{L^4(\Omega)}^2\|\nabla c_{\epsilon}\|_{L^4(\Omega)}^2+2C_S^2\overline{n}_0^2\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2\end{align*}

for all $t\in(0,\infty)$ . Since the Poincaré inequality and the interpolation entail that

\begin{equation*} \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 = \|n_\epsilon-\overline{n_\epsilon}\|_{L^2(\Omega)}^2 \le C_1\|\nabla n_\epsilon\|_{L^2(\Omega)}^2 \end{equation*}

and

\begin{equation*}\|n_\epsilon-\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^2 \le 2\left(\|n_\epsilon\|_{L^1(\Omega)}^{\frac95}\|n_\epsilon\|_{L^\infty(\Omega)}^{\frac15} + \|\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^2\right) \le C_2 \left(\|n_0\|_{L^1(\Omega)}^{\frac95} + \|n_0\|_{L^1(\Omega)}^2\right),\end{equation*}

we deduce from the Gagliardo–Nirenberg inequality, the Young inequality, Lemma 3.8 and the interpolation that

\begin{align*}&2C_S^2 \|n_\epsilon-\overline{n}_0\|_{L^4(\Omega)}^2\|\nabla c_{\epsilon}\|_{L^4(\Omega)}^2 \\[3pt]&\le C_3\left(\|\nabla(n_\epsilon-\overline{n}_0)\|_{L^2(\Omega)}^{\frac54}\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{\frac34}+\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^2\right)\left(\|\nabla c_{\epsilon}\|_{L^{14}(\Omega)}^{\frac7{6}}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac5{6}} \right) \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ C_4\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{2}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac{20}9}+C_4\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{2}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac56} \\[3pt] &\le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ C_5\|n_\epsilon-\overline{n}_0\|_{L^{\frac32}(\Omega)}^{2}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac56} \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ C_6\|n_\epsilon-\overline{n}_0\|_{L^{2}(\Omega)}^{\frac76}\|n_\epsilon-\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^{\frac56}\|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^{\frac56} \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2+ \frac{1}{4C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + C_7 \|n_\epsilon-\overline{n}_0\|_{L^{\frac{10}9}(\Omega)}^2 \|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^2 \\[3pt] & \le \frac12\|\nabla n_{\epsilon}\|_{L^2(\Omega)}^2 + \frac{1}{4C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + C_8 \left(\|n_0\|_{L^1(\Omega)}^{\frac95} + \|n_0\|_{L^1(\Omega)}^2 \right) \|\nabla c_{\epsilon}\|_{L^{2}(\Omega)}^2\end{align*}

and thus that

(4.11) \begin{align}\frac{d}{dt}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +\frac{1}{4 C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2& \le C_9 \left(\|n_0\|_{L^1(\Omega)}^{\frac95} + \|n_0\|_{L^1(\Omega)}^2\right)\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2 \nonumber \\& \le C_9 \left(\delta^{\frac95} + \delta^2\right)\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2\le \frac{1}{4C_1}\|\nabla c_{\epsilon}\|_{L^2(\Omega)}^2\end{align}

for all $t\in(0,\infty)$ provided that $\delta$ is suitable small.

To absorb the right-hand side of (4.11), we test equation (1.9) $_2$ by $c_{\epsilon}-\overline{n}_0$ , integrate by parts over $\Omega$ and utilise the Hölder inequality and the Young inequality to obtain

\begin{equation*}\frac{\epsilon}{2}\frac{d}{dt}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 + \|\nabla c_\epsilon\|_{L^2(\Omega)}^2+\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 \le \frac12\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2+ \frac12\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation*}

for all $t\in(0,\infty)$ , and then have

(4.12) \begin{equation}\epsilon\frac{d}{dt}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +2 \|\nabla c_\epsilon\|_{L^2(\Omega)}^2+\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation}

for all $t\in(0,\infty)$ . This together with (4.11) yields that

\begin{equation*}\frac{d}{dt}\Big(\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2+\frac{\epsilon}{ 8C_1}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\Big)+\frac{1}{ 8C_1}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +\frac{1}{ 8C_1}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le 0\end{equation*}

for all $t\in(0,\infty)$ . Without loss of generality, we can assume $0<\epsilon<\min\big\{1, 8C_1\big\}$ and set

\begin{equation*}y(t):=\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2+\frac{\epsilon}{ 8C_1}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation*}

to obtain

(4.13) \begin{equation}y'(t)+\frac{1}{ 8C_1}y(t)\le 0 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation}

Integrating (4.13) from 0 to t, we conclude that

\begin{equation*}y(t)\le e^{-\frac{t}{ 8C_1}}y(0)\le \Big(\|n_0-\overline{n}_0\|_{L^2(\Omega)}^2+\frac{1}{ 8C_1}\|c_0-\overline{n}_0\|_{L^2(\Omega)}^2\Big)e^{-\frac{t}{8C_1}}\le C_{10} e^{-\frac{t}{ 8C_1}}\end{equation*}

for all $t\in(0,\infty)$ , which entails (4.10).

Lemma 4.7. Under the assumption of Lemma 4.6, there exist two positive constants C and $\mu$ such that

\begin{equation*}\|c_{\epsilon}(\cdot,t)-\overline{n}_0\|_{L^2({\Omega})}\leq Ce^{-\mu t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

provided that $\epsilon$ is suitable small.

Proof. By repeating the proof of (4.12), we can see from Lemma 4.6 that

\begin{equation*}\epsilon\frac{d}{dt}\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2 +2 \|\nabla c_\epsilon\|_{L^2(\Omega)}^2+\|c_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\le C_1e^{-\mu_1t}\end{equation*}

for all $t\in(0,\infty)$ and some positive constants $C_1$ and $\mu_1$ . Then by setting

\begin{equation*}y(t):=\|c_\epsilon(\cdot,t)-\overline{n}_0\|_{L^2(\Omega)}^2\end{equation*}

and letting $0<\epsilon<\min\big\{1, \frac1{2\mu_1}\big\}$ , we have

\begin{equation*}\epsilon y'(t) + y(t) \le C_1e^{-\mu_1t}\end{equation*}

and thus

\begin{equation*}y(t)\le y(0)e^{-\frac{t}{\epsilon}} + \frac{C_1e^{-\frac{t}{\epsilon}}}{\epsilon} \int_0^t e^{(\frac{1}{\epsilon}-\mu_1)s}ds\le y(0)e^{-\frac{t}{\epsilon}} + 2C_1e^{-\mu_1t}\le C_2 e^{-\min\{\mu_1, 1\}t}\end{equation*}

for all $t\in(0,\infty)$ . This completes the proof of Lemma 4.7.

Corollary 4.1. Under the assumptions of Lemma 4.6, there exist two positive constants C and $\mu$ such that for any $p>1$ ,

\begin{equation*}\|c_{\epsilon}(\cdot,t)-\overline{n}_0\|_{W^{1,\,p}({\Omega})} \le Ce^{-\mu t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

provided that $\epsilon$ is suitable small.

Proof. For any fixed $q>p>1$ , we use the Gagliardo–Nirenberg inequality, the $W^{1,q}$ boundedness of $c_{\epsilon}$ and Lemma 4.7 to get

\begin{align*}\|c_{\epsilon}-\overline{n}_0\|_{W^{1,\,p}({\Omega})}& \le C_1\|c_{\epsilon}-\overline{n}_0\|^{\theta_1}_{W^{1,q}({\Omega})}\|c_{\epsilon}-\overline{n}_0\|^{1-\theta_1}_{L^{2}({\Omega})}+C_1\|c_{\epsilon}-\overline{n}_0\|_{L^{2}({\Omega})} \\[2pt]& \le C_1\big( \|c_{\epsilon}\|_{W^{1,q}({\Omega})} + \|\overline{n}_0\|_{W^{1,q}({\Omega})}\big)^{\theta_1} \|c_{\epsilon}-\overline{n}_0\|^{1-\theta_1}_{L^{2}({\Omega})}+C_1\|c_{\epsilon}-\overline{n}_0\|_{L^{2}({\Omega})} \\[2pt] & \le C_2 \|c_{\epsilon}-\overline{n}_0\|^{1-\theta_1}_{L^{2}({\Omega})} + C_1\|c_{\epsilon}-\overline{n}_0\|_{L^{2}({\Omega})} \\[2pt] & \le C_3e^{-\mu t}\end{align*}

for some $\mu>0$ and all $t\in(0,\infty)$ , where $ \theta_1 = \frac{q(p-1)}{(q-1)p}\in(0,1)$ .

Lemma 4.8. Under the assumptions of Lemma 4.6, there exist two positive constants C and $\mu$ such that

(4.14) \begin{equation}\|n_{\epsilon}(\cdot,t)-\overline{n}_0\|_{L^{\infty}({\Omega})}\leq Ce^{-\mu t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation}

and

(4.15) \begin{equation}\|u_\epsilon(\cdot,t)\|_{L^{\infty}({\Omega})}\leq Ce^{-\mu t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation}

provided that $\epsilon$ is suitable small.

Proof. Let $\big(e^{t\Delta}\big)_{t\ge0}$ be the homogeneous Neumann heat semigroup in $\Omega$ . Since

\begin{equation*}n_{\epsilon}(\cdot,t)=e^{t\triangle}n_0-\int_0^t e^{(t-s)\triangle}\Big(\nabla\cdot\big(n_{\epsilon} S(\cdot,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)+u_\epsilon\cdot\nabla n_{\epsilon}\Big)(\cdot,s)ds\end{equation*}

for all $t\in(0,\infty)$ , we can use the fact $e^{t\triangle}\overline{n}_0=\overline{n}_0$ to obtain

(4.16) \begin{align}\|n_{\epsilon}(\cdot,t)-\overline{n}_0\|_{L^{\infty}({\Omega})} &\le \|e^{t\triangle}(n_0-\overline{n}_0)\|_{L^{\infty}({\Omega})}+\int_0^t \|e^{(t-s)\triangle}\nabla\cdot\big(n_{\epsilon} S(\cdot,n_{\epsilon},c_{\epsilon})\nabla c_{\epsilon}\big)(\cdot,s)\|_{L^{\infty}({\Omega})}ds \nonumber\\[3pt] &\qquad +\int_0^t \|e^{(t-s)\triangle}u_\epsilon(\cdot,s)\cdot\nabla n_{\epsilon}(\cdot,s)\|_{L^{\infty}({\Omega})}ds \nonumber\\[3pt] &:= J_1 + J_2 + J_3, \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{align}

To estimate $J_1$ , $J_2$ and $J_3$ , we first deduce from the asymptotics of the Neumann heat semigroup (see Lemma 1.3 in [Reference Winkler35]) that

\begin{equation*}J_1\le k_1 e^{-\lambda_1 t} \|n_0-\overline{n}_0\|_{L^{\infty}({\Omega})}\le C_1 e^{-\lambda_1 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

due to $\int_{\Omega}(n_0-\overline{n}_0)dx=0$ ,

\begin{equation*}J_2\le k_4C_S\int_0^t\Big(1+(t-s)^{-\frac12-\frac13}\Big)e^{-\lambda_1(t-s)}\|n_{\epsilon}(\cdot,s)\cdot\nabla c_{\epsilon}(\cdot,s)\|_{L^{3}({\Omega})}ds \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

and

\begin{align*}J_3& = \int_0^t \|e^{(t-s)\triangle}\nabla\cdot \big((n_{\epsilon}-\overline{n}_0)u_\epsilon\big)(\cdot,s)\|_{L^{\infty}({\Omega})}ds \\& \le k_4\int_0^t\Big(1+(t-s)^{-\frac12-\frac1{q}}\Big)e^{-\lambda_1(t-s)}\|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|_{L^{q}({\Omega})}ds \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align*}

for any fixed $2<q<3$ with some positive constants $k_1, k_4$ and $\lambda_1$ . Then since Corollaries 3.2 and 4.1 entail that

\begin{align*}\|n_{\epsilon}(\cdot,s)\cdot\nabla c_{\epsilon}(\cdot,s)\|_{L^{3}({\Omega})}&\le \|n_{\epsilon}(\cdot,s)\|_{L^{4}({\Omega})}\|\nabla c_{\epsilon}(\cdot,s)\|_{L^{12}({\Omega})} \\[3pt] & =\|n_{\epsilon}(\cdot,s)\|_{L^{4}({\Omega})}\|\nabla \big(c_{\epsilon}(\cdot,s)-\overline{n}_0\big)\|_{L^{12}({\Omega})}\le C_2 e^{-\mu_1 s}\end{align*}

for some $\mu_1>0$ , we can further estimate $J_2$ as

\begin{equation*}J_2\le C_3\int_0^t\Big(1+(t-s)^{-\frac56}\Big)e^{-\lambda_1(t-s)} e^{-\mu_1 s}ds\le C_4 e^{-\min\{\lambda_1, \mu_1\} t}\end{equation*}

for all $t\in(0,\infty)$ . Next, we can take a similar procedure to further estimate $J_3$ . Indeed, it follows from the interpolation, Lemmas 4.6, 3.5 and Corollary 3.2 that

\begin{align*}& \|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|_{L^{q}({\Omega})} \\[3pt] &\le \|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|^{\frac{3-q}{q}}_{L^{\frac32}({\Omega})}\|(n_{\epsilon}(\cdot,s)-\overline{n}_0)u_\epsilon(\cdot,s)\|^{\frac{2q-3}{q}}_{L^{3}({\Omega})}\\[3pt] &\le C_5\Big(\|n_{\epsilon}(\cdot,s)-\overline{n}_0\|_{L^{2}({\Omega})} \|u_\epsilon(\cdot,s)\|_{L^{6}({\Omega})} \Big)^{\frac{3-q}{q}} \Big(\|n_{\epsilon}(\cdot,s)-\overline{n}_0\|_{L^{4}({\Omega})} \|u_\epsilon(\cdot,s)\|_{L^{12}({\Omega})} \Big)^{\frac{2q-3}{q}} \\[3pt] & \le C_6 e^{-\mu_2 s}\end{align*}

for some $\mu_2>0$ , which entails that

\begin{equation*}J_3\le C_7\int_0^t\Big(1+(t-s)^{-\frac12-\frac1{q}}\Big)e^{-\lambda_1(t-s)} e^{-\mu_2 s}\le C_8 e^{-\min\{\lambda_1, \mu_2\} t}\end{equation*}

for all $t\in(0,\infty)$ . Inserting $J_1, J_2$ and $J_3$ into (4.16), we can achieve (4.14).

In order to get (4.15), it is sufficient to show that

\begin{equation*}\|A^\beta u_\epsilon(\cdot,t)\|_{L^{2}({\Omega})}\le Ce^{-\mu t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

with $\beta\in(\frac12,1)$ and $\mu>0$ due to the embedding $D(A^\beta)\hookrightarrow L^\infty(\Omega)$ . For this purpose, we first test equation (1.9) $_3$ by $u_\epsilon$ , integrate by parts over $\Omega$ and employ the solenoidality of $u_\epsilon$ , the Hölder inequality, Lemma 4.6 and Lemma 3.5 to obtain that

\begin{align*}\frac{1}{2} \frac{d}{dt} \|u_\epsilon\|_{L^2(\Omega)}^2 + \|\nabla u_\epsilon\|_{L^2(\Omega)}^2&= \int_{\Omega}n_{\epsilon}\nabla\phi\cdot u_\epsilon \\[3pt] &=\int_{\Omega}(n_{\epsilon}-\overline{n}_0)\nabla\phi\cdot u_\epsilon \\[3pt] &\le \|\nabla \phi\|_{L^\infty(\Omega)}\|n_{\epsilon}-\overline{n}_0\|_{L^2(\Omega)}\|u_\epsilon\|_{L^2(\Omega)}\le C_9 e^{-\mu_3 t}\end{align*}

for all $t\in(0,\infty)$ with some $\mu_3>0$ . Due to $u_\epsilon=0$ on $\partial\Omega$ , it then follows from the Poincaré inequality that

\begin{equation*}\frac{d}{dt} \|u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2+ \frac{1}{C_{10}}\| u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2 \le 2C_9 e^{-\mu_3 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Thus, without loss of generality, we may assume $\mu_3<\frac{1}{C_{10}}$ to deduce that

(4.17) \begin{align}\|u_\epsilon(\cdot,t)\|_{L^2(\Omega)}^2 \le e^{-\frac{t}{C_{10}}}\|u_0\|_{L^2(\Omega)}^2+C_{11} e^{-\mu_3 t}\le C_{12}e^{-\mu_3 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{align}

Following the proof of (3.13), we have

\begin{align*}& \frac12\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2+\|A u_\epsilon\|_{L^2(\Omega)}^2 \\[3pt] &=-\int_\Omega \mathcal{P}(u_\epsilon\cdot\nabla u_\epsilon)\cdot Au_\epsilon+\int_\Omega \mathcal{P}(n_\epsilon\nabla \phi)\cdot Au_\epsilon\\[3pt] &=-\int_\Omega \mathcal{P}(u_\epsilon\cdot\nabla u_\epsilon)\cdot Au_\epsilon+\int_\Omega \mathcal{P}\big((n_\epsilon-\overline{n}_0)\nabla \phi\big)\cdot Au_\epsilon\\[3pt] & \le C_{13} \Big(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\Big)\Big(\|A u_\epsilon\|_{L^2(\Omega)}^{\frac12}\|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\|\nabla u_\epsilon\|_{L^2(\Omega)}\Big)\|A u_\epsilon\|_{L^2(\Omega)} \\[3pt] &\qquad + C_{13} \|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}\|A u_\epsilon\|_{L^2(\Omega)}\\[3pt] & \le \frac12\|A u_\epsilon\|_{L^2(\Omega)}^2+C_{14} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{2}\left( \left(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\right)^4 \right.\\[3pt] & \qquad \left.+\left(\| u_\epsilon\|_{L^2(\Omega)}^{\frac12} \|\nabla u_\epsilon\|_{L^2(\Omega)}^{\frac12}+\| u_\epsilon\|_{L^2(\Omega)}\right)^2 \right)+C_{14}\|n_\epsilon-\overline{n}_0\|_{L^2(\Omega)}^2\end{align*}

for all $t\in(0,\infty)$ , which together with (4.17), Lemmas 4.6 and 3.4 yields that

\begin{equation*}\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2+\|A u_\epsilon\|_{L^2(\Omega)}^2 \le C_{15}e^{-\mu_4 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

with some $\mu_4>0$ . Since

\begin{equation*}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2\le C_{16}\|A u_\epsilon\|_{L^2(\Omega)}\| u_\epsilon\|_{L^2(\Omega)}+C_{16}\| u_\epsilon\|_{L^2(\Omega)}^2\le \|A u_\epsilon\|_{L^2(\Omega)}^2 + C_{17}\| u_\epsilon\|_{L^2(\Omega)}^2\end{equation*}

for all $t\in(0,\infty)$ , we have

\begin{equation*}\frac{d}{dt}\|\nabla u_\epsilon\|_{L^2(\Omega)}^2 + \|\nabla u_\epsilon\|_{L^2(\Omega)}^2 \le C_{15}e^{-\mu_4 t} + C_{17}\| u_\epsilon\|_{L^2(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

This together with (4.17) warrants that

(4.18) \begin{align}\|\nabla u_\epsilon\|_{L^2(\Omega)} \le C_{18}e^{-\mu_5 t} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align}

with some $\mu_5>0$ . Noticing that $\mathcal{P}\big( \overline{n}_0 \nabla\phi \big)=0$ , we can use the variation-of-constants representation

\begin{equation*} u_\epsilon(\cdot, t) = e^{-tA}u_0 + \int^t_0 e^{-(t-s){A}}\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s) ds\end{equation*}

of $u_\epsilon$ to get

\begin{align*}\|A^\beta u_\epsilon\|_{L^2(\Omega)} &\le \|A^\beta e^{-tA}u_0\|_{L^2(\Omega)}+\int^t_0 \|{A}^{\beta} e^{-(t-s){A}}\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s) \|_{L^2(\Omega)}ds \\&\le K_1t^{-\beta}e^{-\mu_6 t}\|u_0\|_{L^2(\Omega)}\\[3pt] &\quad +K_1\int^t_0 (t-s)^{-\beta}e^{-\mu_6 (t-s)}\|\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s)\|_{L^2(\Omega)}ds\end{align*}

for all $t\in(0,\infty)$ with some $K_1>0$ and $\mu_6>0$ by Lemma 2.3(i) in [Reference Yu, Wang and Zheng40]. Since Lemma 3.9, (4.18) and Lemma 4.6 entail that

\begin{align*}& \|\mathcal{P}\big(u_\epsilon\cdot\nabla u_\epsilon - (n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s)\|_{L^2(\Omega)} \\[3pt]& \le \|(u_\epsilon\cdot\nabla u_\epsilon) (\cdot,s)\|_{L^2(\Omega)} + \|(n_{\epsilon}-\overline{n}_0)\nabla\phi \big)(\cdot,s)\|_{L^2(\Omega)} \\[3pt] & \le C_{19} \| u_\epsilon\|_{L^\infty(\Omega)}\|\nabla u_\epsilon\|_{L^2(\Omega)}+C_{19}\| n_{\epsilon}-\overline{n}_0\|_{L^2(\Omega)}\|\nabla\phi\|_{L^\infty(\Omega)} \\[3pt] & \le C_{20}e^{-\mu_7 s} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align*}

with some $\mu_7>0$ , we have

(4.19) \begin{align}\|A^\beta u_\epsilon\|_{L^2(\Omega)}&\le K_1t^{-\beta}e^{-\mu_6 t}\|u_0\|_{L^2(\Omega)} + K_1C_{20}\int^t_0 (t-s)^{-\beta}e^{-\mu_6 (t-s)}e^{-\mu_7 s}ds \nonumber \\[3pt] &\le C_{21} e^{-\mu_6 t} + C_{22} e^{-\mu_7 t} \nonumber \\[3pt] & \le C_{23} e^{-\mu_8 t} \qquad \mathrm{for\,\, all}\quad t\in(1,\infty)\end{align}

with some $\mu_8>0$ . On the other hand, it is clear that

\begin{equation*}\|A^\beta u_\epsilon\|_{L^2(\Omega)}\le \|A^\beta u_0\|_{L^2(\Omega)} + C_{22} e^{-\mu_7 t} \le C_{24} e^{-\mu_8 t} \qquad \mathrm{for\,\, all}\quad t\in(0,1],\end{equation*}

which together with (4.19) implies (4.15). This completes the proof of Lemma 4.8.

We now apply the exponential decay estimate of $(n_{\epsilon}, c_{\epsilon}, u_\epsilon)$ to establish an at most $\frac12$ -order growth on time t in the convergence of the PP-fluid system (1.9) to the PE-fluid system (1.10).

Lemma 4.9. Under the assumptions of Lemma 4.6, there exists $C>0$ such that for each suitable small $\epsilon\in(0,1)$ ,

\begin{equation*}\|\widehat{n}(\cdot,t)\|_{L^2(\Omega)}+\|\widehat{u}(\cdot,t)\|_{L^2(\Omega)} \le C(1+t)^{\frac12} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{equation*}

and

\begin{equation*}\|\widehat{n}\|_{L^2((0,t);\ W^{1, 2}(\Omega))}+\|\widehat{c}\|_{L^2((0,t);\ W^{1, 2}(\Omega))} + \|\widehat{u}\|_{L^2((0,t);\ W^{1, 2}(\Omega))}\le C(1+t)^{\frac12} \epsilon^{\frac12} \quad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. We will deduce our desired result by analysing an entropy-like evolution estimate involving $\widehat{n}$ , $\widehat{u}$ and $c_{\epsilon}$ of the form

\begin{equation*}K\|\widehat{n} (\cdot,t)\|_{L^2(\Omega)}^2 + \epsilon\|c_{\epsilon}(\cdot,t)\|_{L^{2}(\Omega)}^{2}+\|\widehat{u} (\cdot,t)\|_{L^2(\Omega)}^2\end{equation*}

with some K to be determined. For this purpose, testing equation (4.1) $_1$ by $\widehat{n}$ and integrating by parts over $\Omega$ , we obtain

\begin{align*} & \frac{1}{2}\frac{d}{dt} \|\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 \nonumber \\[3pt] &= \int_{\Omega} \Big( (n-\overline{n}_0)\widehat{u} + \widehat{n}S(x,n_{\epsilon},c_{\epsilon}) \nabla(c_{\epsilon}-\overline{n}_0) + (n-\overline{n}_0)S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} + \overline{n}_0S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} \Big) \cdot\nabla\widehat{n} \nonumber\\[3pt] \ &\qquad + \int_{\Omega} n\big(S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c)\big) \cdot \nabla (c-\overline{n}_0)\cdot\nabla\widehat{n} \nonumber\\[3pt] \end{align*}
\begin{align*}\\[-40pt] &\le { \frac14}\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^2 + C_1 \|n-\overline{n}_0\|_{L^{\infty}(\Omega)}^2\|\widehat{u}\|_{L^{2}(\Omega)}^2 + C_1 \|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\|\widehat{n}\|_{L^{4}(\Omega)}^2 \nonumber \\[3pt] & \quad + C_1 \|n-\overline{n}_0\|_{L^{\infty}(\Omega)}^2\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 + C_1 \overline{n}_0^2\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2\\[3pt] &\quad + C_1 \int_{\Omega}n^2 \big|S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c) \big|^2 |\nabla (c-\overline{n}_0)|^2.\end{align*}

Noticing that the Gagliardo–Nirenberg inequality entails that

\begin{align*} &C_1 \|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\|\widehat{n}\|_{L^{4}(\Omega)}^2\\[3pt] &\le C_2\|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\left(\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^{\frac12}\|\widehat{n}\|_{L^{2}(\Omega)}^{\frac12}+\|\widehat{n}\|_{L^{2}(\Omega)}\right)^2\\[3pt] &\le \frac18\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^{2}+C_3\left(\|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^4+\|\nabla(c_{\epsilon}-\overline{n}_0)\|_{L^{4}(\Omega)}^2\right)\|\widehat{n}\|_{L^{2}(\Omega)}^2\end{align*}

and

\begin{align*} & C_1 \int_{\Omega}n^2 \big|S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c) \big|^2 |\nabla (c-\overline{n}_0)|^2 \\&\le 2 C_1 \int_{\Omega}n^2 \left( \big|S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c_{\epsilon})\big|^2 + \big|S(x,n,c_{\epsilon})-S(x,n,c)\big|^2 \right) |\nabla (c-\overline{n}_0)|^2 \nonumber \\&\le 2 C_1 \int_{\Omega}n^2 \left( \big| \nabla S(x,\xi,c_{\epsilon})\big|^2 |\widehat{n}|^2 + \big| \nabla S(x,n, \eta) \big|^2 |\widehat{c} |^2 \right) |\nabla (c-\overline{n}_0)|^2 \nonumber \\& \le C_4 \|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^2 \left( \|\widehat{n}\|_{L^{4}(\Omega)}^2+ \|\widehat{c}\|_{L^4(\Omega)}^2\right)\\& \le \frac18\|\nabla\widehat{n}\|_{L^{2}(\Omega)}^{2} + \gamma\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 {+} C_5\left(\!\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^4+\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^2\right)\left(\!\|\widehat{n}\|_{L^{2}(\Omega)}^{2}+\|\widehat{c}\|_{L^{2}(\Omega)}^{2}\right)\!,\end{align*}

where $\xi$ lies between $n_{\epsilon}$ and n, and $\eta$ lies between $c_{\epsilon}$ and c, respectively, while $\gamma>0$ is to be determined, we can use Corollary 4.1 and Lemma 4.8 to deduce that

(4.20) \begin{align} & \frac{d}{dt} \|\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 \nonumber \\& \le 2\gamma\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 + C_6\big(e^{-\mu_1t} + \bar{n}_0^2\big)\|\nabla\widehat{c}\|_{L^{2}(\Omega)}^2 + C_6 e^{-\mu_1t}\|\widehat{c}\|_{L^{2}(\Omega)}^2 + C_6 e^{-\mu_1t}\left(\|\widehat{n}\|_{L^{2}(\Omega)}^2+\|\widehat{u}\|_{L^{2}(\Omega)}^2\right)\end{align}

for all $t\in(0,\infty)$ with some $\mu_1>0$ .

We next test equation (4.1) $_2$ by $\widehat{c}$ and use the integration by parts over $\Omega$ to obtain

\begin{align*}& \frac{\epsilon}{2}\frac{d}{dt} \|c_{\epsilon}\|_{L^2(\Omega)}^2 + \|\nabla \widehat{c}\|_{L^2(\Omega)}^2+ \|\widehat{c}\|_{L^2(\Omega)}^2 \\& = \epsilon\int_{\Omega}\partial_tc_{\epsilon} c +\int_{\Omega} \big(\widehat{n} - \widehat{u}\cdot\nabla (c-\overline{n}_0) \big) \widehat{c} \nonumber\\ & \le \epsilon\int_{\Omega}\partial_tc_{\epsilon} c + \frac12\|\widehat{c}\|_{L^{2}(\Omega)}^2 + \|\widehat{n}\|_{L^{2}(\Omega)}^2 + \|\nabla (c-\overline{n}_0)\|_{L^{4}(\Omega)}^2\|\widehat{u}\|_{L^{4}(\Omega)}^2 \\ &\le \epsilon\int_{\Omega}\partial_tc_{\epsilon} c + \frac12\|\widehat{c}\|_{L^{2}(\Omega)}^2 + \|\widehat{n}\|_{L^{2}(\Omega)}^2 + \frac14\|\nabla\widehat{u}\|_{L^{2}(\Omega)}^2 + C_7\Big(\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^4\\[2pt] &\quad +\|\nabla(c-\overline{n}_0)\|_{L^{4}(\Omega)}^2\Big) \|\widehat{u}\|_{L^{2}(\Omega)}^2.\end{align*}

Due to $\int_{\Omega}\widehat{n}=0$ , the Poincaré inequality and Corollary 4.1 imply that

(4.21) \begin{align}& \epsilon\frac{d}{dt} \|c_{\epsilon}\|_{L^2(\Omega)}^2 + 2\|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \|\widehat{c}\|_{L^2(\Omega)}^2 \nonumber \\[3pt]& \le 2\epsilon\int_{\Omega}\partial_tc_{\epsilon} c + \frac12\|\nabla\widehat{u}\|_{L^{2}(\Omega)}^2 + C_8 \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + C_8 e^{-\mu_2t}\|\widehat{u}\|_{L^2(\Omega)}^2\end{align}

for all $t\in(0,\infty)$ with some $\mu_2>0$ .

Finally, multiplying equation (4.1) $_3$ by $\widehat{u}$ and using the Poincaré inequality again since $\widehat{u}=0$ on $\partial\Omega$ and $\int_{\Omega}\widehat{n}=0$ , we see that

\begin{align*} \frac{1}{2}\frac{d}{dt} \|\widehat{u}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 =\int_{\Omega} \widehat{n} \nabla\phi \cdot \widehat{u}& \le \|\nabla\phi\|_{L^{\infty}(\Omega)}\|\widehat{u}\|_{L^2(\Omega)}\|\widehat{n}\|_{L^2(\Omega)} \nonumber\\ & \le C_9 \|\nabla \widehat{u}\|_{L^2(\Omega)} \|\nabla\widehat{n}\|_{L^2(\Omega)} \le \frac1{2}\|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + C_{10} \|\nabla\widehat{n}\|_{L^2(\Omega)}^2\end{align*}

and thus that

(4.22) \begin{equation} \frac{d}{dt} \|\widehat{u}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2\le C_{11} \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 \qquad \text{for all} \quad t\in(0,\infty).\end{equation}

We now take $T_1$ large enough such that

\begin{equation*}C_6 e^{-\mu_1t}<\frac{1}{ 6(C_8 + C_{11})} \qquad \text{for all} \quad t\in(T_1, \infty)\end{equation*}

and $\delta$ suitable small such that

\begin{equation*} C_6\overline{n}^2_0<\frac{1}{ 6(C_8 + C_{11})}. \end{equation*}

Then by setting

\begin{equation*} \gamma=\frac{1}{12(C_8 + C_{11})} \qquad \text{and} \qquad K= 2 (C_8+ C_{11}),\end{equation*}

we deduce from (4.20)–(4.22) that

\begin{align*} & \frac{d}{dt}\Big(K\|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) + \frac{K}2 \|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \frac12 \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + \frac23 \|\widehat{c}\|_{L^2(\Omega)}^2 \nonumber\\ &\le 2\epsilon \int_{\Omega}\partial_tc_{\epsilon} c + C_{12} e^{-\mu t} \Big(\|\widehat{n}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) \qquad \quad \text{for all} \quad t\in(T_1, \infty)\end{align*}

with $\mu=\min\big\{\mu_1, \mu_2\big\}$ , which together with the Poincaré inequality yields that

\begin{align*} & \frac{d}{dt}\Big(K \|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) + C_{13} \Big( K \|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2 \Big) \nonumber\\ &\qquad + \frac14\Big( K\|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + \|\widehat{c}\|_{L^2(\Omega)}^2 \Big) \nonumber\\ &\le C_{14} e^{-\mu t}\Big(K\|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\Big) + C_{13} \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+ 2\epsilon \int_{\Omega}\partial_tc_{\epsilon} c.\end{align*}

For simplicity, we set

\begin{equation*}y(t):= K \|\widehat{n}\|_{L^2(\Omega)}^2 + \epsilon \|c_{\epsilon}\|_{L^2(\Omega)}^2+\|\widehat{u}\|_{L^2(\Omega)}^2\end{equation*}

and

\begin{equation*}g(t):= \frac14\Big( K\|\nabla\widehat{n}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{c}\|_{L^2(\Omega)}^2 + \|\nabla\widehat{u}\|_{L^2(\Omega)}^2 + \|\widehat{c}\|_{L^2(\Omega)}^2 \Big)\end{equation*}

as well as

\begin{align*}h(t):= C_{13} \|c_{\epsilon}\|_{L^2(\Omega)}^2+ 2 \int_{\Omega}\partial_tc_{\epsilon} c,\end{align*}

we have

\begin{equation*}y'(t) + C_{13} y(t) + g(t)\le C_{14} e^{-\mu t}y(t) + \epsilon h(t) \qquad \quad \text{for all} \quad t\in(T_1, \infty).\end{equation*}

If we further take $T_2$ large enough such that $2 C_{14} e^{-\mu t} < C_{13}$ for all $t>T_2$ , we can conclude that

\begin{equation*}y'(t) + \frac{ C_{13}}{2}y(t) + g(t) \le \epsilon h(t) \qquad \mathrm{for\,\, all}\quad t\in(T_0,\infty)\end{equation*}

with $T_0:=\max\big\{T_1, T_2\big\}$ . It then follows from Lemmas 4.1, 4.2 and the uniform boundedness of $c_{\epsilon}$ that

\begin{align*}y(t)&\le y(T_0)+ \epsilon \int^t_{T_0} h(s)ds = y(T_0) + \epsilon\left( C_{13} \int^t_{T_0}\|c_{\epsilon}\|_{L^2(\Omega)}^2ds+2 \int^t_{T_0}\int_{\Omega}\partial_tc_{\epsilon} c ds\right) \nonumber\\[3pt] &\le \left((K+1) C_{15} e^{C_{15} T_0}\epsilon+\epsilon\|c_{\epsilon}(\cdot, T_0)\|_{L^2(\Omega)}^2\right) + \epsilon C_{15} (1+t):= C_{16} (1+t)\epsilon\end{align*}

for all $t\in(T_0,\infty)$ , which also implies that

\begin{equation*}\int^t_{T_0} g(s)ds\le y(T_0) + \epsilon \int^t_{T_0}h(s)ds \le C_{16} (1+t)\epsilon \qquad \mathrm{for\,\, all}\quad t\in(T_0,\infty).\end{equation*}

This together with the Poincaré inequality yields the desired results on $(T_0,\infty)$ , while on $(0, T_0)$ , the conclusions follow from Lemma 4.2 directly. This completes the proof of Lemma 4.9.

Applying the second estimate in Lemma 4.9 to the first inequality in (4.6), we can also obtain the growth estimate of $\nabla\widehat{u}$ .

Corollary 4.2. Under the assumptions of Lemma 4.6, there exists $C>0$ such that for each suitable small $\epsilon\in(0,1)$ ,

\begin{equation*}\|\nabla\widehat{u}(\cdot,t)\|_{L^{2}(\Omega)}\le C(1+t)^{\frac12} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Lemma 4.10. Under the assumptions of Lemma 4.6, for any given $\theta\in\big(\frac12,\frac34\big)$ , there exists a positive constant $C(\theta)$ such that for each suitable small $\epsilon\in(0,1)$ ,

\begin{equation*}\|A^\theta\widehat{u}(\cdot,t)\|_{L^2(\Omega)} \le C(\theta) (1+t)^{\frac34} \epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. Similar to the proof of Lemma 4.4, we can see from Lemma 4.9, Corollary 4.2 and (4.9) that

\begin{align*}\|{A}^{\theta}\widehat{u}(\cdot,t)\|_{L^2(\Omega)}& \le C_1 \int^t_0 (t-s)^{-\theta}e^{-\lambda(t-s)} \Big( \|\nabla\widehat{u}\|_{L^2(\Omega)} + \|\widehat{u}\|_{L^4(\Omega)}\|\nabla{u}\|_{L^4(\Omega)} + \|\widehat{n}\|_{L^2(\Omega)}\Big)ds \\&\le C_2\epsilon^{\frac12}\int^t_0(1+s)^{\frac12}(t-s)^{-\theta}e^{-\lambda(t-s)} \big( 1+ \|\nabla{u}\|_{L^4(\Omega)} \big) ds\\&\le C_3(1+t)^{\frac12}\epsilon^{\frac12} + C_3(1+t)^{\frac12}\epsilon^{\frac12}\left(\int^t_0\big((t-s)^{-\theta}e^{-\lambda(t-s)}\big)^\frac43 ds\right)^{\frac34}\\&\quad \times \left(\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds\right)^{\frac14}\\&\le C_4(1+t)^{\frac12}\epsilon^{\frac12} +C_4(1+t)^{\frac12}\epsilon^{\frac12}\left(\int^t_0\|\nabla{u}(\cdot,s)\|_{L^4(\Omega)}^4 ds\right)^{\frac14}\\&\le C_5(1+t)^{\frac34}\epsilon^{\frac12} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty). \qquad \\[-40pt] \end{align*}

Lemma 4.11. Under the assumptions of Lemma 4.6, for any given $ p>2$ , there exists a positive constant C(p) such that for each suitable small $\epsilon\in(0,1)$ ,

\begin{equation*} \|\widehat{n}(\cdot,t)\|_{L^p(\Omega)}\le C(p)(1+t)^{\frac{1}{2}} \epsilon^{\frac{2}{p^2}} \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Proof. We first multiply equation (4.1) $_1$ by $\widehat{n}^{\,p-1}$ with $p>2$ and integrate by parts over $\Omega$ to obtain

\begin{align*} & \frac{1}{p}\frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{4(p-1)}{p^2}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 \nonumber \\[3pt] & = (p-1) \int_{\Omega} \Big( (n-\overline{n}_0)\widehat{n}^{\,p-2}\widehat{u} + \widehat{n}^{\,p-1}S(x,n_{\epsilon},c_{\epsilon}) \nabla(c_{\epsilon}-\overline{n}_0) + (n-\overline{n}_0) \widehat{n}^{\,p-2} S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} \nonumber \\[3pt] & \qquad + \overline{n}_0 \widehat{n}^{\,p-2} S(x,n_{\epsilon},c_{\epsilon}) \nabla\widehat{c} \Big) \cdot\nabla\widehat{n} +(p-1) \int_{\Omega} n \widehat{n}^{\,p-2} \big(S(x,n_{\epsilon},c_{\epsilon})-S(x,n,c)\big) \nabla (c-\overline{n}_0)\cdot\nabla\widehat{n} \nonumber \\[3pt] & \le \frac{p-1}4 \int_{\Omega} \widehat{n}^{\,p-2}|\nabla\widehat{n}|^2 + C_1 \|n-\overline{n}_0\|_{L^\infty(\Omega)}^2\int_{\Omega}\widehat{n}^{\,p-2} |\widehat{u}|^2 + C_1 \int_{\Omega}\widehat{n}^p |\nabla(c_{\epsilon}-\overline{n}_0)|^2 \nonumber \\[3pt] & \qquad + C_1 \|n-\overline{n}_0\|_{L^\infty(\Omega)}^2\int_{\Omega}\widehat{n}^{\,p-2} |\nabla\widehat{c}|^2 + C_1 \overline{n}_0\int_{\Omega}\widehat{n}^{\,p-2}|\nabla\widehat{c}|^2 \\[3pt] & \qquad + C_1\|n\|_{L^{\infty}(\Omega)}^2 \int_{\Omega} \widehat{n}^{\,p-2} \big(\widehat{n}^{\,2} + \widehat{c}^{\,2}\big) |\nabla(c-\overline{n}_0)|^2. \end{align*}

Noting that Lemmas 4.8, 3.9 and Corollary 4.1 entail that

\begin{equation*}\|n-\overline{n}_0\|_{L^\infty(\Omega)}^2 \le C_2 e^{-\mu t} \quad \mathrm{and}\quad \|\nabla(c-\overline{n}_0)\|_{L^q(\Omega)}^2 \le C_2 e^{-\mu t}\end{equation*}

for all $t\in(0,\infty)$ and any $q>1$ with some $\mu>0$ , we can use the Hölder inequality, the boundedness of $n_{\epsilon}$ and n, Corollary 4.1 and the Young inequality to obtain that for any fixed $q_0>p$ ,

\begin{align*}\\[-40pt] & \int_{\Omega} \widehat{n}^{\,p-2} \left(\widehat{n}^{\,2} + \widehat{c}^{\,2}\right) |\nabla(c-\overline{n}_0)|^2 \\[3pt]&\le \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \|\widehat{n}\|_{L^{q_0}(\Omega)}^{p-2} \|\nabla(c-\overline{n}_0)\|_{L^{\frac{2pq_0}{(p-2)(q_0-p)}}(\Omega)}^2\\[3pt] &\le \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \|\widehat{n}\|_{L^{p}(\Omega)}^{\frac{p(p-2)}{q_0}}\|n_{\epsilon}-n\|_{L^{\infty}(\Omega)}^{\frac{(p-2)(q_0-p)}{q_0}}\|\nabla(c-\overline{n}_0)\|_{L^{\frac{2pq_0}{(p-2)(q_0-p)}}(\Omega)}^2\\[3pt] &\le C_3 e^{-\mu t} \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \|\widehat{n}\|_{L^{p}(\Omega)}^{\frac{p(p-2)}{q_0}}\\[3pt] &\le C_4 e^{-\mu t} \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right) \left(\|\widehat{n}\|_{L^{p}(\Omega)}^{p-2}+1\right)\\[3pt] &\le C_5 e^{-\mu t}\left(\|\widehat{n}\|_{L^{p}(\Omega)}^p + \|\widehat{c}\|_{L^{p}(\Omega)}^p\right) + C_5e^{-\mu t} \left(\|\widehat{n}\|_{L^{p}(\Omega)}^2 + \|\widehat{c}\|_{L^{p}(\Omega)}^2\right).\end{align*}

Similarly, we have

\begin{equation*}\int_{\Omega}\widehat{n}^p |\nabla(c_{\epsilon}-\overline{n}_0)|^2\le C_6e^{-\mu t}\|\widehat{n}\|_{L^{p}(\Omega)}^p + C_6e^{-\mu t}\|\widehat{n}\|_{L^{p}(\Omega)}^2\end{equation*}

and thus deduce from the Young inequality that

\begin{align*} & \frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{3(p-1)}{p} \|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 \nonumber \\[3pt] & \le C_7 e^{-\mu t} \Big(\|\widehat{n}\|_{L^p(\Omega)}^p + \|\widehat{u}\|_{L^p(\Omega)}^p\Big) + C_7 \overline{n}_0 \|\widehat{n}\|_{L^{p}(\Omega)}^{p} + C_7 \big( e^{-\mu t} + \overline{n}_0\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^{p}\nonumber \\[3pt] & \qquad + C_7e^{-\mu t}\|\widehat{n}\|_{L^{p}(\Omega)}^2+C_7e^{-\mu t}\|\widehat{c}\|_{L^{p}(\Omega)}^2. \end{align*}

Without loss of generality, we assume $ C_7>1$ and take suitable small $\delta$ such that $\overline{n}_0<1$ . Then since

\begin{align*}\|\widehat{n}\|_{L^p(\Omega)}^p = \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2& \le C_8\left(\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^{\frac{2(p-2)}{p}} \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{4}{p}} + \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^2\right) \\[3pt] &\le \frac{p-1}{ C_7 p}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + C_9 \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^2 \end{align*}

and

\begin{equation*} \begin{split}\|\widehat{n}\|_{L^p(\Omega)}^2 =\|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^{\frac4{p}}& \le C_{10}\left(\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^{\frac{p-2}{p}} \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{2}{p}} + \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}\right)^{\frac4{p}}\\[3pt] &\le \frac{p-1}{2C_7p}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + C_{11}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{8}{p^2-2p+4}}+C_{11}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{4}{p}} \end{split} \end{equation*}

we can use the fact

\begin{equation*} \|\widehat{u}\|_{L^p(\Omega)}^p \le C_{12}\Big(\|\widehat{u}\|_{L^2(\Omega)}^2\|\nabla\widehat{u}\|_{L^2(\Omega)}^{p-2}+\|\widehat{u}\|_{L^2(\Omega)}^p \Big), \end{equation*}

Lemma 4.9 and Corollary 4.2 to obtain

\begin{align*} & \frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac12 \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 \nonumber \\ & \le \frac{d}{dt} \|\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{p-1}{2p}\|\nabla\widehat{n}^{\,\frac{p}2}\|_{L^2(\Omega)}^2 + \frac{C_9}{2} \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^2 \nonumber \\ & \le \left( C_7C_9 e^{-\mu t} + C_7 C_9 + \frac{C_9}{2}\right) \|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4p}(\Omega)}^2 + C_7 e^{-\mu t} \|\widehat{u}\|_{L^p(\Omega)}^p + C_7 \big( e^{-\mu t} + 1\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^{p} \nonumber \\ & \quad + C_7C_{11}e^{-\mu t}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{8}{p^2-2p+4}}+C_7C_{11}e^{-\mu t}\|\widehat{n}^{\,\frac{p}2}\|_{L^{\frac4{p}}(\Omega)}^{\frac{4}{p}}+C_7e^{-\mu t}\|\widehat{c}\|_{L^p(\Omega)}^2 \nonumber \\ & \le C_{13} (1+t)^{\frac{p}2}\epsilon^{\frac{p}2} + C_{13}(1+t)^{\frac{2p}{p^2-2p+4}}\epsilon^{\frac{2p}{p^2-2p+4}}+C_{13}(1+t)\epsilon \nonumber \\ & \qquad +C_7 \big( e^{-\mu t} + 1\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^p+ C_7 e^{-\mu t}\|\widehat{c}\|_{L^p(\Omega)}^2 \nonumber \\ &\le 3C_{13}(1+t)^{\frac{p}2}\epsilon^{\frac{2p}{p^2-2p+4}} + C_7 \big( e^{-\mu t} + 1\big) \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^p + C_7 e^{-\mu t}\|\widehat{c}\|_{L^p(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty) \end{align*}

due to $p>2$ and $\epsilon\in(0,1)$ . By setting

\begin{equation*}y(t):=\|\widehat{n}(\cdot,t)\|_{L^p(\Omega)}^p =\|\widehat{n}^{\,\frac{p}2}(\cdot,t)\|_{L^2(\Omega)}^2,\end{equation*}

we have

\begin{equation*}y'(t) + \frac12y(t)\le C_{14} (1+t)^{\frac{p}2}\epsilon^{\frac{2p}{p^2-2p+4}} + C_{14} \|\widehat{c}\|_{W^{1,\, p}(\Omega)}^p + C_{14}\|\widehat{c}\|_{L^p(\Omega)}^2 \qquad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{equation*}

Thus, a direct calculation implies that

(4.23) \begin{equation}y(t) \le C_{14} \epsilon^{\frac{2p}{p^2-2p+4}} \int_0^t (1+s)^{\frac{p}2}e^{ -\frac{(t-s)}{2}} ds + C_{14} \int_0^t e^{ -\frac{(t-s)}{2}} \left(\|\widehat{c}(\cdot, s)\|_{W^{1,\, p}(\Omega)}^{p} + \|\widehat{c}(\cdot, s)\|_{L^p(\Omega)}^2 \right)ds \end{equation}

for all $t\in(0,\infty)$ due to $y(0)=0$ . For the first integral on the right-hand side of (4.23), we have

\begin{equation*} \int_0^t (1+s)^{\frac{p}2}e^{-\frac{(t-s)}{2}} ds\le (1+t)^{\frac{p}2}\int_0^t e^{-\frac{(t-s)}{2}} ds \le (1+t)^{\frac{p}2},\end{equation*}

while for the second one, we can make use of the interpolation inequality, Lemmas 3.8, 4.9 and the Hölder inequality to deduce that

\begin{align*}\int_0^t e^{ -\frac{(t-s)}{2}} \|\widehat{c}(\cdot, s)\|_{W^{1,\, p}(\Omega)}^{p} ds& \le C_{15} \int_0^t e^{ -\frac{(t-s)}{2}} \|\widehat{c}(\cdot, s)\|_{W^{1, 2}(\Omega)}^{\frac{2(q-p)}{q-2}} \|\widehat{c}(\cdot, s)\|_{W^{1, q}(\Omega)}^{\frac{q(p-2)}{q-2}} ds \nonumber \\&\le C_{16} \int_0^t e^{ -\frac{(t-s)}{2}} \|\widehat{c}(\cdot, s)\|_{W^{1, 2}(\Omega)}^{\frac{2(q-p)}{q-2}} ds \nonumber \\&\le C_{17} \left(\int_0^t \|\widehat{c}(\cdot, s)\|_{W^{1, 2}(\Omega)}^{2} ds\right)^{\frac{q-p}{q-2}} \left(\int_0^t e^{-\frac{q-2}{p-2}\frac{(t-s)}{2}} ds\right)^{\frac{p-2}{q-2}} \nonumber \\&\le C_{18} (1+t)^{\frac{q-p}{q-2}}\epsilon^{\frac{q-p}{q-2}} \qquad \quad \mathrm{for\,\, all}\quad t\in(0,\infty)\end{align*}

with $q>p+2>4$ . Similarly, for the third integral, we also have

\begin{align*}\int_0^t e^{-\frac{(t-s)}{2}}\|\widehat{c}(\cdot, s)\|_{L^{ p}(\Omega)}^{2} ds& \le C_{19} \int_0^t e^{-\frac{(t-s)}{2}}\|\widehat{c}(\cdot, s)\|_{L^{ 2}(\Omega)}^{\frac{4}{p}} \|\widehat{c}(\cdot, s)\|_{L^{\infty}(\Omega)}^{\frac{2(p-2)}{p}} ds \nonumber \\&\le C_{20} \left(\int_0^t \|\widehat{c}(\cdot, s)\|_{L^{ 2}(\Omega)}^{2} ds\right)^{\frac{2}{p}} \left(\int_0^t e^{-\frac{p}{p-2}\frac{(t-s)}{2}} ds\right)^{\frac{p-2}{p}} \nonumber \\&\le C_{21} (1+t)^{\frac{2}{p}}\epsilon^{\frac{2}{p}} \qquad \quad \mathrm{for\,\, all}\quad t\in(0,\infty).\end{align*}

These three estimates together with (4.23) yield that

\begin{equation*}y(t)\le C_{22} (1+t)^{\frac{p}2}\epsilon^{\frac{2p}{p^2-2p+4}} + C_{22} (1+t)^{\frac{q-p}{q-2}} \epsilon^{\frac{q-p}{q-2}} + C_{22} (1+t)^{\frac{2}{p}}\epsilon^{\frac{2}{p}}\le C_{23} (1+t)^{\frac{p}{2}} \epsilon^{\frac{2}{p}}\end{equation*}

for all $ t\in(0,\infty)$ . This completes the proof of Lemma 4.11.

Proof of Theorem 1.2. A direct application of Corollary 4.1 and Lemmas 4.84.11 implies the desired conclusion.

5. Numerical experiments

In this section, we carry out the numerical simulations to look into the convergence behaviour depending on $\epsilon$ and t. Set $\Omega = \{x \in \mathbb{R}^2 : |x| < 1 \}$ . We solve the PP-fluid system (1.9) and the PE-fluid system (1.10) with the chemotactic sensitivity

\begin{equation*}S(x,n,c) = \frac{1}{(1+n)^\alpha}\left[ \begin{array}{c@{\quad}c}a_1 & b_1 \\ \\[-8pt] b_2 & a_2\end{array} \right].\end{equation*}

We will compute and plot the following norms to observe the experimental convergence rates:

\begin{equation*}\begin{aligned}& \mathrm{nL2}=\|\widehat{n}(t)\|_{L^2}/\|n(t)\|_{L^2}, && \quad \mathrm{nL4}= \|\widehat{n}(t)\|_{L^4}/\|n(t)\|_{L^4}, && \quad \mathrm{nH1}=\|\widehat{n}(t)\|_{H^1}/\|n(t)\|_{H^1}, \\[3pt] & \mathrm{cH1}=\|\widehat{c}(t)\|_{H^1}/\| c(t) \|_{H^1}, && \quad \mathrm{uLInf}=\|\widehat{u}(t)\|_{L^\infty}/\|u(t)\|_{L^\infty}, && \quad \mathrm{uH1}=\|\widehat{u}(t)\|_{H^1}/\|u(t)\|_{H^1}.\end{aligned}\end{equation*}

In all numerical examples, we take the initial velocity $u_0 = (0,0)$ and assign the initial concentration $c_0$ as the solution of

\begin{equation*}-\Delta c_0 + c_0 = n_0 \quad \text{ in } \Omega, \quad \quad \nabla c_0 \cdot \nu = 0 \quad \text{ on } \partial \Omega.\end{equation*}

We select three initial cell densities $n_0$ (presented below), which result in three different types of solution.

Figure 1. Example 1. (n, c, u) of the PE-fluid system.

Figure 2. Example 1. $(n_\epsilon,c_\epsilon,u_\epsilon)$ of the PP-fluid system with $\epsilon=0.5$ .

Example 1. We set $\alpha =\frac14$ , $a_1 = a_2 = b_1 = b_2 = \frac12$ , $\nabla\phi = (1,-1)$ . The initial density $n_0$ is given by:

\begin{equation*} n_0= 30\left(e^{-5\left|x-\left(\frac{1}{2},-\frac{1}{2}\right)\right|^2 } + e^{-5\left|x-\left(-\frac{1}{2},-\frac{1}{2}\right)\right|^2}\right) + 20\left(e^{-5\left|x-\left(\frac{1}{2},\frac{1}{2}\right)\right|^2} + e^{-5\left|x-\left(-\frac{1}{2},\frac{1}{2}\right)\right|^2}\right).\end{equation*}

The evolution of the PE-fluid system and PP-fluid system with $\epsilon = 0.5$ is shown in Figures 1 and 2, respectively, where the colour bars illustrate the magnitude of $(n,c,|u|)$ and $(n_\epsilon,c_\epsilon,|u_\epsilon|)$ , and the arrows in the figures of u and $u_\epsilon$ represent the direction of velocity. Both two solutions tend to the nontrivial equilibrium with the cells and chemical concentrating at the lower left quarter, whereas the PE-fluid system approaches the equilibrium quicker, and the motion of the PP-fluid system seems to be a delay of the PE-fluid system. In the following, we investigate the convergence of $(\widehat{n}, \widehat{c}, \widehat{u})$ depending on $\epsilon$ and t, respectively.

Now let us fix $t=2$ and compute the norms of $(\widehat{n}(t),\widehat{c}(t),\widehat{u}(t))$ on different $\epsilon$ . The results are plotted in log-scale in Figure 3 (Ex1- $\epsilon$ ). All the errors decrease in the same rate to the solid straight line $y=\epsilon$ , which means that the experiment enjoys the $O(\epsilon)$ -convergence. It is better than the theoretical results (Theorems 1.1 and 1.2), but not a surprise because any sufficiently smooth solution (in particular, the norms of $\partial_t c_\epsilon$ are bounded independent of the reciprocal of $\epsilon$ ) will result the optimal convergence.

Figure 3. We fix $t=2$ for Example 1, $t=0.3$ for Example 2 and $t=4$ for Example 3. The three figures, (Ex1- $\epsilon$ ), (Ex2- $\epsilon$ ) and (Ex3- $\epsilon$ ), show the norms of $(\widehat{n}(t), \widehat{c}(t), \widehat{u}(t))$ depending on $\epsilon$ for the Example 1, 2 and 3, respectively. The errors plotted in log-scale decrease as with $\epsilon$ (the solid straight line), which indicates the convergence rate $O(\epsilon)$ for all three numerical examples.

Next, we fix $\epsilon = 2^{-5}$ and compute the errors for various t (see Figure 4 (Ex1-t)). Be aware of the PE-fluid system and PP-fluid system tend to the same nontrivial equilibrium, the deviation $(\widehat{n},\widehat{c},\widehat{u})$ will extinguish. Figure 4 (Ex1-t) exhibits the decreasing of the error over t. However, it is saturated near $10^{-6}$ due to the numerical approximation.

Figure 4. We fix $\epsilon=2^{-5}$ . The three figures, (Ex1-t), (Ex2-t) and (Ex3-t), display the norms of $(\widehat{n}, \widehat{c}, \widehat{u})$ on various t for the Example 1, 2 and 3, respectively. We plot the errors in log-scale.

Example 2. We replace the initial value $n_0$ of the previous example by a symmetry function:

\begin{equation*}\begin{aligned}& n_0= 30\left(e^{-5\left|x-\left(\frac{1}{2},-\frac{1}{2}\right)\right|^2 } + e^{-5\left|x-\left(-\frac{1}{2},\frac{1}{2}\right)\right|^2}\right) + 20\left(e^{-5\left|x-\left(\frac{1}{2},\frac{1}{2}\right)\right|^2} + e^{-5\left|x-\left(-\frac{1}{2},-\frac{1}{2}\right)\right|^2}\right).\end{aligned}\end{equation*}

Both (n, c) and $(n_\epsilon, c_\epsilon)$ rapidly tend to the constant equilibriums $(\overline{n}_0,\overline{n}_0)$ (see Figures 5 and 6). The difference between the PE-fluid system and PP-fluid system is tiny. In this example, the velocity u and $u_\epsilon$ vanish quickly. In fact, the amplitudes of u and $u_\epsilon$ are very small (see the colour bar of u and $u_\epsilon$ ), and the pressures P and $P_\epsilon$ tend to the function $\overline{n}_0 \phi = \overline{n}_0(x-y)$ almost instantly (note that $\nabla \phi = (1,-1)$ and $\nabla P = n \nabla \phi$ when $u = 0$ ).

Figure 5. Example 2. (n, c, u) of the PE-fluid system.

Figure 6. Example 2. $(n_\epsilon,c_\epsilon,u_\epsilon)$ of the PP-fluid system with $\epsilon=0.25$ .

Figure 7. Example 3. (n, c, u) of the PE-fluid system.

Figure 8. Example 3. $(n_\epsilon,c_\epsilon,u_\epsilon)$ of the PP-fluid system with $\epsilon=0.5$ .

Now we fix $t=0.3$ and plot the error for various $\epsilon$ (see Figure 3 (Ex2- $\epsilon$ )), which reveals the $O(\epsilon)$ -convergence of the fast signal diffusion limit. Next for fixed $\epsilon = 2^{-5}$ , we compute and plot the error for different t in Figure 4 (Ex2-t). Since both the PE-fluid system and PP-fluid system tend to the trivial equilibrium $(\overline{n}_0,\overline{n}_0,0,\overline{n}_0 \phi)$ , the norms of $(\widehat{n},\widehat{c},\widehat{u})$ will diminish to 0. The experimental errors of n and c decrease over t but are bounded below because the numerical error exists. We mention that the experimental $\|\widehat{u}(t)\|_{L^\infty}$ and $\|\widehat{u}(t)\|_{H^1}$ also decrease to 0 in the simulation. But the computation results of the relative errors $\mathrm{uLInf}=\|\widehat{u}(t)\|_{L^\infty}/\|u(t)\|_{L^\infty}$ and $\mathrm{uH1}=\|\widehat{u}(t)\|_{H^1}/\|u(t)\|_{H^1}$ are polluted because the denominators also tend to zero. Therefore, we shall ignore the curves of uLInf and uH1 in (Ex2-t), which cannot demonstrate the real convergence behaviour of $\|\widehat{u}\|$ .

Example 3. Change the parameters $\alpha = \frac12$ , $a_1 = a_2 = 2$ , $-b_1 = b_2 = 1$ of the chemotactic sensitivity function and take the initial density

\begin{equation*}\begin{aligned}n_0= 60\left(e^{-10|x|^2 } + 20e^{-5\left|x-\left(-\frac{1}{2},-\frac{1}{2}\right)\right|^2}\right).\end{aligned}\end{equation*}

We carry out the simulation with the anticlockwise rotating aggregations (see Figures 7 and 8). The cells and chemicals first disperse and move to the lower left quarter, and then gather and rotate anticlockwise along the boundary. The circular motion of the aggregation is nonuniform. Roughly speaking, the rotation accelerates at the lower left quarter and decelerates near the upper right quarter. The PP-fluid system with smaller $\epsilon$ has a shorter rotation period and the solution is closer to the PE-fluid system.

Figure 3. (Ex3- $\epsilon$ ) indicates the $O(\epsilon)$ -convergence of $(\widehat{n}(t),\widehat{c}(t),\widehat{u}(t))$ for fixed $t=4$ . Figure 4 (Ex3-t) shows the errors on t for fixed $\epsilon = 2^{-5}$ . When the rotation of the PE-fluid system is tardy near $t=4$ and $t=14$ , the evolution of the PP-fluid system catches up and the difference becomes small. On the contrary, the rotation of the PE-fluid system speeds up around $t=10$ and $t=20$ , while motion of the PP-fluid system is delayed, which causes the big increment of error.

Remark 5.1. Theorem 1.1 concludes the $O\left(e^{Ct} \epsilon^\frac{1}{2}\right)$ -convergence for the general global bounded solution, and Theorem 1.2 improves the estimate to $O\left(t^{\frac{1}{2}} \epsilon^\frac{1}{2}\right)$ for the solutions with the trivial equilibrium. Above three numerical examples further explore this topic by investigating the experimental convergence behaviour on different types of solutions. If $\partial_t c_\epsilon$ is smooth enough and bounded independent of $\epsilon^{-\ell}$ ( $\ell>0$ ), it seems quite possible to obtain the $O(\epsilon)$ -convergence, and the numerical results confirm that. When the PE-fluid system and PP-fluid system tend to the same equilibrium, nontrivial or trivial, the deviation between them shall vanish as t increases, which is revealed in the Example 1 and 2. The case of the rotating solution is complicated and interesting. The error between the PE-fluid system and PP-fluid system fluctuates drastically in consequence of the nonuniform rotating speed together with the different rotation periods, which brings challenges to the elaborate convergence analysis.

Acknowledgements

The authors are very grateful to all three referees for their detailed comments and valuable suggestions, which greatly improved the manuscript. Z. Xiang was partially supported by the NNSF of China (no. 11971093), the Applied Fundamental Research Program of Sichuan Province (no. 2020YJ0264) and the Fundamental Research Funds for the Central Universities (no. ZYGX2019J096). G. Zhou was partially supported by the NNSF of China (nos. 12171071, 12071061).

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